{"entry_id": "95998eb2-0751-4dd8-b735-6c9a4594b762", "infos": [{"txt": "Adaptive Locally Linear Embedding", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 0, "bbox": [[0.295, 0.121, 0.705, 0.142]]}]}, "tags": ["title"], "label": "article_title", "web_segment_id": 7, "global_sentence_id": 0, "edu_l1_label": "BOT"}, {"txt": "Ali Goli1, Mahdieh Alizadeh1, and Hadi Sadoghi Yazdi∗1,2", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 100000, "bbox": [[0.269, 0.162, 0.73, 0.171]]}]}, "tags": ["text"], "label": "author", "web_segment_id": 10, "global_sentence_id": 1, "edu_l1_label": "EDU_O"}, {"txt": "1Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 2Center of Excellence in Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 200000, "bbox": [[0.184, 0.188, 0.816, 0.204], [0.14, 0.205, 0.86, 0.219], [0.401, 0.222, 0.599, 0.234]]}]}, "tags": ["text"], "label": "author", "web_segment_id": 8, "global_sentence_id": 2, "edu_l1_label": "EDU_O"}, {"txt": "M", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 300000, "bbox": [[0.027, 0.273, 0.054, 0.287]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 13, "global_sentence_id": 3, "edu_l1_label": "EDU_O"}, {"txt": "April 10, 2025", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 400000, "bbox": [[0.441, 0.26, 0.559, 0.275]]}]}, "tags": ["text"], "label": "publish_time", "web_segment_id": 11, "global_sentence_id": 4, "edu_l1_label": "EDU_O"}, {"txt": "20", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 500000, "bbox": [[0.027, 0.298, 0.059, 0.327]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 14, "global_sentence_id": 5, "edu_l1_label": "EDU_O"}, {"txt": "Abstract", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.467, 0.321, 0.533, 0.332]]}]}, "tags": ["title"], "label": "abstract", "web_segment_id": 9, "global_sentence_id": 6, "edu_l1_label": "BOS"}, {"txt": "Manifold learning techniques, such as Locally linear embedding (LLE), are designed to preserve the local neighborhood structures of high-dimensional data during dimensionality reduction.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.202, 0.341, 0.82, 0.352], [0.18, 0.355, 0.786, 0.366]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 7, "edu_l1_label": "IOS"}, {"txt": "Traditional LLE employs Euclidean distance to defne neighborhoods, which can struggle to capture the intrinsic geometric relationships within complex data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.786, 0.355, 0.82, 0.366], [0.18, 0.368, 0.82, 0.38], [0.18, 0.382, 0.57, 0.394]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 8, "edu_l1_label": "IOS"}, {"txt": "A novel approach, Adaptive locally linear embedding(ALLE), is introduced to address this limitation by incorporating a dynamic, data-driven metric that enhances topological preservation.This method redefnes the concept of proximity by focusing on topological neighborhood inclusion rather than fxed distances.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.57, 0.382, 0.821, 0.394], [0.18, 0.396, 0.82, 0.407], [0.18, 0.41, 0.82, 0.421], [0.18, 0.424, 0.792, 0.435]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 9, "edu_l1_label": "IOS"}, {"txt": "By adapting the metric based on the local structure of the data, it achieves superior neighborhood preservation, particularly for datasets with complex geometries and high-dimensional structures.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.792, 0.424, 0.82, 0.435], [0.18, 0.438, 0.82, 0.449], [0.18, 0.451, 0.82, 0.463]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 10, "edu_l1_label": "IOS"}, {"txt": "Experimental results demonstrate that ALLE signifcantly improves the alignment between neighborhoods in the input and feature spaces, resulting in more accurate and topologically faithful embeddings.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.18, 0.465, 0.82, 0.477], [0.18, 0.479, 0.82, 0.49], [0.18, 0.493, 0.261, 0.504]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 11, "edu_l1_label": "IOS"}, {"txt": "This approach advances manifold learning by tailoring distance metrics to the underlying data, providing a robust solution for capturing intricate relationships in high-dimensional datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.261, 0.493, 0.82, 0.504], [0.18, 0.507, 0.82, 0.518], [0.18, 0.521, 0.238, 0.532]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 12, "edu_l1_label": "IOS"}, {"txt": "Keywords— Manifold Learning, Adaptive Locally Linear Embedding, Dimensionality Reduction, Topolog-ical Preservation, Complex Geometries, High-Dimensional Data, Topological Neighborhood Inclusion, Intrinsic Geometric Relationships", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 600000, "bbox": [[0.163, 0.542, 0.861, 0.553], [0.139, 0.556, 0.861, 0.567], [0.139, 0.57, 0.3, 0.581]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 4, "global_sentence_id": 13, "edu_l1_label": "IOS"}, {"txt": "pr", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 700000, "bbox": [[0.035, 0.326, 0.056, 0.353]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 12, "global_sentence_id": 14, "edu_l1_label": "EDU_O"}, {"txt": "1Introduction", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 800000, "bbox": [[0.139, 0.606, 0.324, 0.624]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 6, "global_sentence_id": 15, "edu_l1_label": "BOS"}, {"txt": "Locally linear embedding(LLE) is a prominent manifold learning technique designed to reduce the dimensionality of high-dimensional datasets while preserving their intrinsic geometric structure.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 900000, "bbox": [[0.139, 0.64, 0.861, 0.652], [0.139, 0.654, 0.696, 0.665]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 16, "edu_l1_label": "IOS"}, {"txt": "Proposed by Roweis and Saul, LLE operates through a systematic process that includes identifying the K-nearest neighbors for each data point, calculating reconstruction weights to express each point as a linear combination of its neighbors, and ultimately generating a low-dimensional representation that retains local relationships [14].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 900000, "bbox": [[0.696, 0.654, 0.861, 0.665], [0.139, 0.668, 0.861, 0.679], [0.139, 0.682, 0.861, 0.693], [0.139, 0.696, 0.761, 0.707]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 17, "edu_l1_label": "IOS"}, {"txt": "However, LLE traditionally relies on fxed distance metrics, such as Euclidean distance, which may inadequately represent complex data distributions and fail to capture nuanced topological relationships.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 900000, "bbox": [[0.761, 0.696, 0.861, 0.707], [0.139, 0.709, 0.861, 0.721], [0.139, 0.723, 0.671, 0.735]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 18, "edu_l1_label": "IOS"}, {"txt": "In response to these limitations, we introduce a novel approach termed Adaptive LLE(ALLE), which integrates a fexible, data-driven metric into the LLE framework.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1000000, "bbox": [[0.163, 0.737, 0.861, 0.748], [0.139, 0.751, 0.58, 0.762]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 19, "edu_l1_label": "IOS"}, {"txt": "This method not only learns the metric concurrently with the embedding process but also adapts to the local structure of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1000000, "bbox": [[0.58, 0.751, 0.861, 0.762], [0.139, 0.765, 0.759, 0.776]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 20, "edu_l1_label": "IOS"}, {"txt": "By employing an adaptive metric, our approach aims to enhance the preservation of neighborhood structures, leading to embeddings that more accurately refect the intricate relationships present in high-dimensional datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1000000, "bbox": [[0.759, 0.765, 0.861, 0.776], [0.139, 0.779, 0.861, 0.79], [0.139, 0.792, 0.827, 0.804]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 21, "edu_l1_label": "IOS"}, {"txt": "The benefts of training a metric alongside LLE include improved neighborhood preservation and more robust embeddings, as the learned metric can dynamically adjust to the local geometry of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1100000, "bbox": [[0.163, 0.806, 0.861, 0.818], [0.139, 0.82, 0.795, 0.831]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 22, "edu_l1_label": "IOS"}, {"txt": "However, a potential drawback is that without adequate training, the metric may not capture the true data structure, leading to suboptimal embeddings.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1100000, "bbox": [[0.795, 0.82, 0.861, 0.831], [0.139, 0.834, 0.861, 0.845], [0.139, 0.848, 0.369, 0.859]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 23, "edu_l1_label": "IOS"}, {"txt": "Previous studies have explored similar concepts, integrating metrics into manifold learning frameworks.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1200000, "bbox": [[0.163, 0.862, 0.861, 0.873]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 24, "edu_l1_label": "IOS"}, {"txt": "For instance, approaches such as RLLE and Kernel LLE have aimed to enhance LLE’s robustness and capacityto capture non-linear structures [3,26].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1200000, "bbox": [[0.139, 0.875, 0.861, 0.887], [0.139, 0.889, 0.393, 0.901]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 25, "edu_l1_label": "IOS"}, {"txt": "These methodologies typically adhere to specifc topological structures", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1200000, "bbox": [[0.393, 0.889, 0.861, 0.901]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 26, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1300000, "bbox": [[0.157, 0.91, 0.427, 0.924]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 27, "edu_l1_label": "EDU_O"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1300000, "bbox": [[0.494, 0.956, 0.507, 0.968]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 28, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1300000, "bbox": [[0.134, 0.07, 0.49, 0.295]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 5, "global_sentence_id": 29, "edu_l1_label": "EDU_O"}, {"txt": "(a) Standard LLE: Entangled embedding due to the inability to simplify complex regions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1400000, "bbox": [[0.135, 0.3, 0.487, 0.33]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 11, "global_sentence_id": 30, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1500000, "bbox": [[0.136, 0.346, 0.493, 0.544]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 6, "global_sentence_id": 31, "edu_l1_label": "EDU_O"}, {"txt": "(c) Standard LLE: Entangled embedding due to the inability to simplify complex regions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.134, 0.55, 0.489, 0.579]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 12, "global_sentence_id": 32, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1700000, "bbox": [[0.513, 0.07, 0.867, 0.295]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 4, "global_sentence_id": 33, "edu_l1_label": "EDU_O"}, {"txt": "(b) Adaptive LLE: Improved separation of data through metric learning in topology space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1800000, "bbox": [[0.513, 0.301, 0.861, 0.33]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 8, "global_sentence_id": 34, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1900000, "bbox": [[0.513, 0.345, 0.864, 0.544]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 2, "global_sentence_id": 35, "edu_l1_label": "EDU_O"}, {"txt": "(d) Adaptive LLE: Improved separation of data through metric learning in topology space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2000000, "bbox": [[0.513, 0.552, 0.864, 0.578]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 10, "global_sentence_id": 36, "edu_l1_label": "EDU_O"}, {"txt": "Figure 1: Comparison of standard LLE and adaptive LLE in handling complex data structures.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2100000, "bbox": [[0.16, 0.59, 0.845, 0.607]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 13, "global_sentence_id": 37, "edu_l1_label": "EDU_O"}, {"txt": "that facilitate the preservation of local neighborhoods.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2200000, "bbox": [[0.139, 0.636, 0.499, 0.647]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 38, "edu_l1_label": "IOS"}, {"txt": "However, our proposed method distinguishes itself by emphasizing the adaptability of the metric based on data-driven principles, thereby ofering a more generalized framework for neighborhood preservation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2200000, "bbox": [[0.499, 0.636, 0.861, 0.647], [0.139, 0.65, 0.861, 0.661], [0.139, 0.664, 0.415, 0.675]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 39, "edu_l1_label": "IOS"}, {"txt": "In this paper, we aim to further develop this concept, providing a detailed account of our methodology and its theoretical underpinnings, while also comparing it against existing techniques in terms of topology preservation and neighborhood accuracy.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2300000, "bbox": [[0.163, 0.677, 0.861, 0.689], [0.139, 0.691, 0.861, 0.703], [0.139, 0.705, 0.412, 0.716]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 40, "edu_l1_label": "IOS"}, {"txt": "Through this investigation, we seek to contribute to the ongoing discourse on manifold learning and metric learning, highlighting the critical interplay between these domains in advancing dimensionality reduction techniques.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2300000, "bbox": [[0.412, 0.705, 0.861, 0.716], [0.139, 0.719, 0.861, 0.73], [0.139, 0.733, 0.469, 0.744]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 41, "edu_l1_label": "IOS"}, {"txt": "The implementation of our proposed methods, along with supplementary materials, is available at GitHub repository.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2300000, "bbox": [[0.469, 0.733, 0.861, 0.744], [0.139, 0.747, 0.531, 0.758]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 42, "edu_l1_label": "IOS"}, {"txt": "1.1 Motivation", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2400000, "bbox": [[0.139, 0.779, 0.167, 0.794], [0.189, 0.779, 0.295, 0.794]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 9, "global_sentence_id": 43, "edu_l1_label": "IOS"}, {"txt": "In LLE, the goal is to reduce the dimensionality of data while preserving its local geometric properties.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2500000, "bbox": [[0.139, 0.804, 0.795, 0.815]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 44, "edu_l1_label": "IOS"}, {"txt": "However, the standard approach often struggles with complex data structures, leading to entangled embeddings that fail to separate certain regions efectively, as illustrated in Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2500000, "bbox": [[0.795, 0.804, 0.861, 0.815], [0.139, 0.818, 0.861, 0.829], [0.139, 0.831, 0.532, 0.843]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 45, "edu_l1_label": "IOS"}, {"txt": "1a.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2500000, "bbox": [[0.532, 0.831, 0.56, 0.843]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 46, "edu_l1_label": "IOS"}, {"txt": "the proposed method learns a metric in the topology space, which allows for bending, stretching, and compressing the data manifold in ways that better align with the underlying data complexity.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2500000, "bbox": [[0.56, 0.831, 0.861, 0.843], [0.139, 0.845, 0.861, 0.857], [0.139, 0.859, 0.42, 0.87]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 47, "edu_l1_label": "IOS"}, {"txt": "This adaptive metric enables the deformation of the topology, such that complex regions of the data space are simplifed, making it easier to distinguish and separate key features.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.163, 0.873, 0.861, 0.884], [0.139, 0.887, 0.652, 0.898]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 48, "edu_l1_label": "IOS"}, {"txt": "As shown in Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.652, 0.887, 0.77, 0.898]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 49, "edu_l1_label": "IOS"}, {"txt": "1b, proposed approach, produces embeddings with signifcantly improved separation and alignment compared to the standard method.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.77, 0.887, 0.861, 0.898], [0.139, 0.901, 0.861, 0.912], [0.139, 0.915, 0.193, 0.926]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 50, "edu_l1_label": "IOS"}, {"txt": "By reducing the complexity of the data manifold, the proposed method not only disentangles the", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.193, 0.915, 0.861, 0.926]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 51, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2700000, "bbox": [[0.495, 0.956, 0.506, 0.966]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 52, "edu_l1_label": "EDU_O"}, {"txt": "output but also enhances the accuracy of downstream tasks such as classifcation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2700000, "bbox": [[0.139, 0.076, 0.677, 0.087]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 53, "edu_l1_label": "IOS"}, {"txt": "As depicted in Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.163, 0.09, 0.288, 0.101]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 54, "edu_l1_label": "IOS"}, {"txt": "1c and Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.288, 0.09, 0.368, 0.101]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 55, "edu_l1_label": "IOS"}, {"txt": "1d, the plots represent a low-dimensional embedding of the Iris dataset [6] obtained using standard LLE and ALLE, respectively.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.368, 0.09, 0.861, 0.101], [0.139, 0.103, 0.491, 0.115]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 56, "edu_l1_label": "IOS"}, {"txt": "In Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.491, 0.103, 0.541, 0.115]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 57, "edu_l1_label": "IOS"}, {"txt": "1c, the 2D representation generated by standard LLE demonstrates that the embeddings of two specifc classes remain entangled, making it challenging to distinguish between them in the reduced-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.541, 0.103, 0.861, 0.115], [0.139, 0.117, 0.861, 0.129], [0.139, 0.131, 0.534, 0.142]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 58, "edu_l1_label": "IOS"}, {"txt": "In contrast, Fig.1d illustrates the results of applying ALLE, where the topology of the data space is adaptively stretched.This transformation signifcantly improves the separability of the classes, efectively resolving the entanglement issue observed in the standard LLE embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2900000, "bbox": [[0.163, 0.145, 0.861, 0.156], [0.139, 0.159, 0.861, 0.17], [0.139, 0.173, 0.635, 0.184]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 59, "edu_l1_label": "IOS"}, {"txt": "The adaptive approach of ALLE, which leverages metric learning to modify the data’s intrinsic geometry, ensures a more discriminative low-dimensional representation while preserving local relationships.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2900000, "bbox": [[0.635, 0.173, 0.861, 0.184], [0.139, 0.186, 0.861, 0.198], [0.139, 0.2, 0.555, 0.212]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 60, "edu_l1_label": "IOS"}, {"txt": "1.2Topological Space and Neighborhoods", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3000000, "bbox": [[0.139, 0.232, 0.555, 0.247]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 11, "global_sentence_id": 61, "edu_l1_label": "IOS"}, {"txt": "In topology, the concept of neighborhoods defnes the idea of ”closeness” without relying on specifc metrics.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3100000, "bbox": [[0.139, 0.256, 0.843, 0.268]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 62, "edu_l1_label": "IOS"}, {"txt": "Aneighborhood includes a point and all other points considered ”close” by inclusion within an open set, ratherthan a predefned distance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3100000, "bbox": [[0.843, 0.256, 0.861, 0.268], [0.139, 0.27, 0.861, 0.281], [0.139, 0.284, 0.322, 0.295]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 63, "edu_l1_label": "IOS"}, {"txt": "This abstract notion of proximity ofers a powerful fexibility as it emphasizes relationships between points in a more generalized manner, independent of fxed metrics.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3100000, "bbox": [[0.322, 0.284, 0.861, 0.295], [0.139, 0.298, 0.726, 0.309]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 64, "edu_l1_label": "IOS"}, {"txt": "A topology on a set X is a collection T of subsets of X, called open sets, that satisfes three conditions:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3200000, "bbox": [[0.134, 0.311, 0.865, 0.351]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 65, "edu_l1_label": "IOS"}, {"txt": " the empty set and the entire set X are in T ; the union of any collection of sets in T is also in T ; and the intersection of any fnite collection of sets in T is also in T [11].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3200000, "bbox": [[0.134, 0.311, 0.865, 0.351]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 66, "edu_l1_label": "IOS"}, {"txt": "A neighborhood of a point x in a topological space X is a set N containing x such that there exists an open set O where x ∈O ⊆N.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3300000, "bbox": [[0.163, 0.353, 0.861, 0.365], [0.139, 0.367, 0.337, 0.378]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 67, "edu_l1_label": "IOS"}, {"txt": "In this context, N contains an open set surrounding x, refecting a topologicalsense of closeness [23].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3300000, "bbox": [[0.337, 0.367, 0.861, 0.378], [0.139, 0.381, 0.285, 0.392]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 68, "edu_l1_label": "IOS"}, {"txt": "1.3The Metric Topology", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3400000, "bbox": [[0.139, 0.412, 0.393, 0.427]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 10, "global_sentence_id": 69, "edu_l1_label": "IOS"}, {"txt": "This section explores one of the fundamental methods of defning a topology on a set, using a metric.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.139, 0.437, 0.797, 0.448]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 70, "edu_l1_label": "IOS"}, {"txt": "A metric introduces the concept of distance between elements of a set, ofering a precise way to understand how ”close”or ”far apart” these elements are.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.797, 0.437, 0.861, 0.448], [0.139, 0.451, 0.861, 0.462], [0.139, 0.465, 0.358, 0.476]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 71, "edu_l1_label": "IOS"}, {"txt": "Topologies derived from metrics serve as the cornerstone of modern analysisand provide an intuitive yet rigorous framework for studying continuity and convergence.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.358, 0.465, 0.861, 0.476], [0.139, 0.479, 0.727, 0.49]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 72, "edu_l1_label": "IOS"}, {"txt": "Formally, a metric on a set X is a function d :", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.163, 0.492, 0.478, 0.504]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 73, "edu_l1_label": "IOS"}, {"txt": " X × X →R that assigns a non-negative real number toevery pair of points x and y in X, representing the distance between them.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.478, 0.492, 0.861, 0.504], [0.139, 0.506, 0.625, 0.518]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 74, "edu_l1_label": "IOS"}, {"txt": "To qualify as a metric, the function d must satisfy three conditions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.625, 0.506, 0.861, 0.518], [0.139, 0.52, 0.345, 0.531]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 75, "edu_l1_label": "IOS"}, {"txt": "The frst is non-negativity, which requires that d(x, y) ≥0, with equality onlyif x = y, ensuring that distinct points always have a positive distance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.345, 0.52, 0.861, 0.531], [0.139, 0.534, 0.62, 0.545]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 76, "edu_l1_label": "IOS"}, {"txt": "The second is symmetry, meaning d(x, y) = d(y, x), refecting the intuitive idea that the distance between two points is independent of their order.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.62, 0.534, 0.861, 0.545], [0.139, 0.548, 0.861, 0.559], [0.139, 0.562, 0.177, 0.573]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 77, "edu_l1_label": "IOS"}, {"txt": "Finally, the triangle inequality states that d(x, z) ≤d(x, y) + d(y, z), ensuring that the direct distancebetween two points is never greater than the sum of the distances when traveling through a third point.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.177, 0.562, 0.861, 0.573], [0.139, 0.575, 0.825, 0.587]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 78, "edu_l1_label": "IOS"}, {"txt": "Using a metric, the notion of an ϵ-ball, denoted Bd(x, ϵ), is defned as the set of all points y in X such thatd(x, y) < ϵ.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3700000, "bbox": [[0.163, 0.589, 0.861, 0.601], [0.139, 0.603, 0.212, 0.614]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 79, "edu_l1_label": "IOS"}, {"txt": "This ϵ-ball can be thought of as a neighborhood around x, containing all points within a specifedradius ϵ.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3700000, "bbox": [[0.212, 0.603, 0.861, 0.614], [0.139, 0.617, 0.195, 0.628]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 80, "edu_l1_label": "IOS"}, {"txt": "By considering these neighborhoods for every point in X, a metric topology emerges.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3700000, "bbox": [[0.195, 0.617, 0.763, 0.628]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 81, "edu_l1_label": "IOS"}, {"txt": "This topologyorganizes the points of X into a structure that refects their proximity, as determined by the metric, providing a natural way to study the relationships between points based on “closeness.” [11]", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3700000, "bbox": [[0.763, 0.617, 0.861, 0.628], [0.139, 0.631, 0.861, 0.642], [0.139, 0.645, 0.681, 0.656]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 82, "edu_l1_label": "IOS"}, {"txt": "1.4Innovation:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.139, 0.676, 0.297, 0.691]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 13, "global_sentence_id": 83, "edu_l1_label": "IOS"}, {"txt": " Need for a Flexible Metric", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.297, 0.676, 0.563, 0.691]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 13, "global_sentence_id": 84, "edu_l1_label": "IOS"}, {"txt": "Traditional metrics, such as Euclidean distance, often fail to adequately represent complex data distributions, particularly in high-dimensional or non-linear datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3900000, "bbox": [[0.139, 0.701, 0.861, 0.712], [0.139, 0.715, 0.495, 0.726]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 85, "edu_l1_label": "IOS"}, {"txt": "A static metric like Euclidean distance assumes uniform scaling across the data, which leads to inaccuracies in neighborhood preservation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3900000, "bbox": [[0.495, 0.715, 0.861, 0.726], [0.139, 0.728, 0.679, 0.74]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 86, "edu_l1_label": "IOS"}, {"txt": "To address this limitation, we propose a data-dependent metric defned as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3900000, "bbox": [[0.679, 0.728, 0.861, 0.74], [0.139, 0.742, 0.453, 0.754]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 87, "edu_l1_label": "IOS"}, {"txt": "dM(x1,x3) = √(x1 + x3)TM(x1 = x3),", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4000000, "bbox": [[0.373, 0.761, 0.625, 0.776]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 88, "edu_l1_label": "IOS"}, {"txt": "where M is a positive semi-defnite matrix learned from the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.139, 0.782, 0.566, 0.793]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 89, "edu_l1_label": "IOS"}, {"txt": "This metric is designed to adapt to the local geometry, providing a more fexible and accurate representation of proximity by refecting both global and local characteristics of the data distribution.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.566, 0.782, 0.861, 0.793], [0.139, 0.795, 0.861, 0.807], [0.139, 0.809, 0.435, 0.821]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 90, "edu_l1_label": "IOS"}, {"txt": "We introduce a novel approach to learning a metric dM(xi, xj) that adapts to the local geometry of the data, ensuring that neighborhood structures are accurately preserved in the feature space.This metric is learned through an iterative optimization process that minimizes the least-squares error of neighborhood reconstruction, resulting in a more faithful representation of distances in both the input and embedded spaces.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.435, 0.809, 0.861, 0.821], [0.139, 0.823, 0.861, 0.834], [0.139, 0.837, 0.861, 0.848], [0.139, 0.851, 0.861, 0.862], [0.139, 0.865, 0.382, 0.876]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 91, "edu_l1_label": "IOS"}, {"txt": "As illustrated in Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.382, 0.865, 0.523, 0.876]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 92, "edu_l1_label": "IOS"}, {"txt": "2, our method introduces a data-dependent metric that better preserves neighborhood structures in the input space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.523, 0.865, 0.861, 0.876], [0.139, 0.878, 0.574, 0.89]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 93, "edu_l1_label": "IOS"}, {"txt": "By letting the data inform the metric, we achieve embeddings that better refect the intrinsic relationships within complex, high-dimensional datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.574, 0.878, 0.861, 0.89], [0.139, 0.892, 0.861, 0.904]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 94, "edu_l1_label": "IOS"}, {"txt": "This adaptive, context-sensitive approach represents a signifcant advancement in manifold learning, leading to more meaningful embeddings and a more accurate capture of the relationships between data points.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.139, 0.906, 0.861, 0.917], [0.139, 0.92, 0.817, 0.931]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 95, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4200000, "bbox": [[0.494, 0.955, 0.507, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 96, "edu_l1_label": "EDU_O"}, {"txt": "1.5Manifold Perspective", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4300000, "bbox": [[0.139, 0.073, 0.393, 0.088]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 11, "global_sentence_id": 97, "edu_l1_label": "IOS"}, {"txt": "A manifold is a topological space that locally resembles Euclidean space, even though its global structure may be complex.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.139, 0.098, 0.861, 0.109], [0.139, 0.111, 0.219, 0.123]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 98, "edu_l1_label": "IOS"}, {"txt": "Manifold learning techniques leverage this local Euclidean structure to better understand the behavior of high-dimensional data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.219, 0.111, 0.861, 0.123], [0.139, 0.125, 0.37, 0.137]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 99, "edu_l1_label": "IOS"}, {"txt": "The challenge lies in preserving the essential geometric properties of the manifold when embedding it in a lower-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.37, 0.125, 0.861, 0.137], [0.139, 0.139, 0.519, 0.15]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 100, "edu_l1_label": "IOS"}, {"txt": "In manifold learning, methods LLE aim to preserve local neighborhoods when mapping high-dimensional data to a lower-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.519, 0.139, 0.861, 0.15], [0.139, 0.153, 0.723, 0.164]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 101, "edu_l1_label": "IOS"}, {"txt": "However, Euclidean distance, which is typically used to defne these neighborhoods, may not faithfully represent the true underlying structure due to its limitations in scaling uniformly across all points.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.723, 0.153, 0.861, 0.164], [0.139, 0.167, 0.861, 0.178], [0.139, 0.181, 0.586, 0.192]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 102, "edu_l1_label": "IOS"}, {"txt": "Our learned metric enhances this process by allowing proximity to be defned in a way that is sensitive to local variations in the data, thereby improving the accuracy of neighborhood preservation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.586, 0.181, 0.861, 0.192], [0.139, 0.194, 0.861, 0.206], [0.139, 0.208, 0.423, 0.22]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 103, "edu_l1_label": "IOS"}, {"txt": "2Related Work", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4500000, "bbox": [[0.139, 0.245, 0.339, 0.263]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 10, "global_sentence_id": 104, "edu_l1_label": "BOS"}, {"txt": "Dimensionality reduction has been a core technique in machine learning and data analysis, with methods such as Principal Component Analysis [8] and Multidimensional Scaling [17] serving as traditional linear embedding approaches.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4600000, "bbox": [[0.139, 0.278, 0.861, 0.29], [0.139, 0.292, 0.861, 0.303], [0.139, 0.306, 0.216, 0.317]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 105, "edu_l1_label": "IOS"}, {"txt": "These methods focus on preserving global structures but fail to capture the non-linear relationships inherent in complex data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4600000, "bbox": [[0.216, 0.306, 0.861, 0.317], [0.139, 0.32, 0.308, 0.331]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 106, "edu_l1_label": "IOS"}, {"txt": "This limitation has motivated the development of manifold learning techniques.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4600000, "bbox": [[0.308, 0.32, 0.84, 0.331]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 107, "edu_l1_label": "IOS"}, {"txt": "Tenenbaum et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4700000, "bbox": [[0.163, 0.334, 0.281, 0.345]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 108, "edu_l1_label": "IOS"}, {"txt": "[16] introduced a global geometric framework for nonlinear dimensionality reduction, capable of preserving the intrinsic geometry of data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4700000, "bbox": [[0.281, 0.334, 0.861, 0.345], [0.139, 0.347, 0.483, 0.359]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 109, "edu_l1_label": "IOS"}, {"txt": "Their approach efciently uncovers nonlinear structures, such as manifolds, using local metric information to infer global relationships.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4700000, "bbox": [[0.483, 0.347, 0.861, 0.359], [0.139, 0.361, 0.652, 0.373]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 110, "edu_l1_label": "IOS"}, {"txt": "One of the most prominent methods is LLE, introduced by Roweis and Saul [14].LLE constructs a low-dimensional representation by preserving local linear relationships, assuming that data lies on or near a non-linear manifold.It maps high-dimensional data to a lower-dimensional space while preserving local geometries.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4800000, "bbox": [[0.163, 0.375, 0.861, 0.386], [0.139, 0.389, 0.861, 0.4], [0.139, 0.403, 0.861, 0.414], [0.139, 0.417, 0.213, 0.428]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 111, "edu_l1_label": "IOS"}, {"txt": "Neighborhood selection, which is the only nonlinear step of the LLE, plays a very important role in the performance of the algorithm.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4900000, "bbox": [[0.163, 0.431, 0.861, 0.442], [0.139, 0.444, 0.338, 0.456]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 112, "edu_l1_label": "IOS"}, {"txt": "If the points are unevenly or sparsely sampled or contaminated by noise, the estimated reconstruction weights of its neighbors cannot refect well the local geometry of the manifold in the embedding space, even leading to a large bias to the embedding result.Most extensions of LLE are concerning the neighborhood selection [4].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4900000, "bbox": [[0.338, 0.444, 0.861, 0.456], [0.139, 0.458, 0.861, 0.469], [0.139, 0.472, 0.861, 0.483], [0.139, 0.486, 0.417, 0.497]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 113, "edu_l1_label": "IOS"}, {"txt": "also, LLE uses Euclidean distance to defne neighborhoods, which can misrepresent the true geometric relationships in more complex or non-uniformly sampled datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4900000, "bbox": [[0.417, 0.486, 0.861, 0.497], [0.139, 0.5, 0.816, 0.511]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 114, "edu_l1_label": "IOS"}, {"txt": "Several enhancements to LLE have been proposed to address its limitations.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.163, 0.514, 0.678, 0.525]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 115, "edu_l1_label": "IOS"}, {"txt": "Pan et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.678, 0.514, 0.755, 0.525]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 116, "edu_l1_label": "IOS"}, {"txt": "[12] introduced Weighted Locally Linear Embedding (WLLE), which incorporates a novel weighted distance measure to improve neighbor selection and reduce redundancy.WLLE addresses challenges posed by noisy, sparse, or weakly connected data by adapting to the local data distribution.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.755, 0.514, 0.861, 0.525], [0.139, 0.527, 0.861, 0.539], [0.139, 0.541, 0.861, 0.553], [0.139, 0.555, 0.511, 0.566]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 117, "edu_l1_label": "IOS"}, {"txt": "The method demonstrates superior manifold learning performance and robustness to parameter changes compared to standard LLE and NLE.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.511, 0.555, 0.861, 0.566], [0.139, 0.569, 0.713, 0.58]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 118, "edu_l1_label": "IOS"}, {"txt": "Additionally, WLLE’spractical applications, such as in face recognition, highlight its ability to outperform traditional methods like KPCA and kernel direct discriminant analysis in certain scenarios.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.713, 0.569, 0.861, 0.58], [0.139, 0.583, 0.861, 0.594], [0.139, 0.597, 0.579, 0.608]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 119, "edu_l1_label": "IOS"}, {"txt": "[22] proposed Kernel Relative Transformation and Relative Transformation to enhance LLE.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5100000, "bbox": [[0.168, 0.61, 0.775, 0.622]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 120, "edu_l1_label": "IOS"}, {"txt": "These transformations aim to improve neighborhood graph construction by reducing the impact of noise, sparsity, and short-circuit edges, ensuring better preservation of the intrinsic structure of data manifolds.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5100000, "bbox": [[0.775, 0.61, 0.861, 0.622], [0.139, 0.624, 0.861, 0.636], [0.139, 0.638, 0.744, 0.649]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 121, "edu_l1_label": "IOS"}, {"txt": "Chang and Yeung [3] introduced RLLE, which incorporates robust statistics to handle outliers efectively, ensuring stable embeddings even in the presence of noisy data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.163, 0.652, 0.861, 0.663], [0.139, 0.666, 0.555, 0.677]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 122, "edu_l1_label": "IOS"}, {"txt": "Kernel LLE [26] extends the original method by introducing non-linear kernel functions, allowing the method to better capture the structure of complex non-linear manifolds.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.555, 0.666, 0.861, 0.677], [0.139, 0.68, 0.861, 0.691], [0.139, 0.693, 0.279, 0.705]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 123, "edu_l1_label": "IOS"}, {"txt": "Further advancing LLE’s capabilities, [18] proposed Isometric LLE (ISO LLE), whichexplicitly enforces geodesic distance preservation during embedding.Unlike standard LLE, which focuses on local linear reconstructions, ISO LLE optimizes for global isometry by incorporating pairwise distance constraints, bridging the gap between LLE’s local emphasis and Isomap’s global geodesic approach.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.279, 0.693, 0.861, 0.705], [0.139, 0.707, 0.861, 0.719], [0.139, 0.721, 0.861, 0.732], [0.139, 0.735, 0.812, 0.746]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 124, "edu_l1_label": "IOS"}, {"txt": "Theseapproaches have improved the applicability of LLE but remain reliant on predefned distance metrics that may not fully refect the complexity of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.812, 0.735, 0.861, 0.746], [0.139, 0.749, 0.861, 0.76], [0.139, 0.763, 0.424, 0.774]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 125, "edu_l1_label": "IOS"}, {"txt": "Alternative manifold learning methods, such as Isomap [16] and Laplacian Eigenmaps [1], aim to address some of these limitations by incorporating geodesic distances and graph-based techniques to preserve both local and global structures.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5300000, "bbox": [[0.163, 0.776, 0.861, 0.788], [0.139, 0.79, 0.861, 0.802], [0.139, 0.804, 0.315, 0.815]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 126, "edu_l1_label": "IOS"}, {"txt": "While efective, these methods still face challenges in preserving true neighborhood relationships, particularly in highly complex or sparsely sampled datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5300000, "bbox": [[0.315, 0.804, 0.861, 0.815], [0.139, 0.818, 0.628, 0.829]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 127, "edu_l1_label": "IOS"}, {"txt": "To overcome the limitations of fxed distance metrics, metric learning has been introduced as a powerful approach for dynamically adapting the distance function based on the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.163, 0.832, 0.861, 0.843], [0.139, 0.846, 0.644, 0.857]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 128, "edu_l1_label": "IOS"}, {"txt": "Xing et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.644, 0.846, 0.723, 0.857]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 129, "edu_l1_label": "IOS"}, {"txt": "[24] proposed a Mahalanobis distance-based method that learns a metric to better refect the geometry of the data, improving clustering performance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.723, 0.846, 0.861, 0.857], [0.139, 0.859, 0.861, 0.871], [0.139, 0.873, 0.293, 0.885]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 130, "edu_l1_label": "IOS"}, {"txt": "Weinberger and Saul’s Large Margin Nearest Neighbor algorithm [21] further demon-strated the efectiveness of adaptive metrics by optimizing neighborhood relationships, achieving signifcant improvements in classifcation tasks.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.293, 0.873, 0.861, 0.885], [0.139, 0.887, 0.861, 0.898], [0.139, 0.901, 0.378, 0.912]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 131, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5500000, "bbox": [[0.494, 0.956, 0.507, 0.968]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 132, "edu_l1_label": "EDU_O"}, {"txt": "Step 1: Initialization", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5600000, "bbox": [[0.155, 0.231, 0.3, 0.245]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 11, "global_sentence_id": 133, "edu_l1_label": "EDU_O"}, {"txt": "Step 2: Optimization Problem", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5700000, "bbox": [[0.536, 0.246, 0.744, 0.259]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 10, "global_sentence_id": 134, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5800000, "bbox": [[0.144, 0.073, 0.857, 0.262]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 6, "global_sentence_id": 135, "edu_l1_label": "EDU_O"}, {"txt": "Figure 2: Overview of the proposed method.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5900000, "bbox": [[0.339, 0.281, 0.66, 0.298]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 9, "global_sentence_id": 136, "edu_l1_label": "EDU_O"}, {"txt": "Building on this foundation, recent works have explored the integration of metric learning into manifold learning techniques.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.163, 0.327, 0.861, 0.339], [0.139, 0.341, 0.269, 0.352]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 137, "edu_l1_label": "IOS"}, {"txt": "Xu and Li [25] introduced Robust Locally Nonlinear Embedding, which constructs nonlinear relationships between local neighborhoods using basis functions and constrained least squares.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.269, 0.341, 0.861, 0.352], [0.139, 0.355, 0.772, 0.366]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 138, "edu_l1_label": "IOS"}, {"txt": "This method aligns neighborhoods with locally nonlinear patches of the manifold, enhancing robustness to noise and outperforming conventional LLE-based methods.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.772, 0.355, 0.861, 0.366], [0.139, 0.369, 0.861, 0.38], [0.139, 0.383, 0.419, 0.394]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 139, "edu_l1_label": "IOS"}, {"txt": "Liu et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.419, 0.383, 0.487, 0.394]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 140, "edu_l1_label": "IOS"}, {"txt": "[10] proposed the LMR-LLE algorithm, which improves traditional LLE by incorporating multiple neighborhood structures through local mutual representation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.487, 0.383, 0.861, 0.394], [0.139, 0.396, 0.825, 0.408]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 141, "edu_l1_label": "IOS"}, {"txt": "This approach fuses diverse local structures into a global reconstruction model, achieving superior performance in tasks such as bearing fault diagnosis.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6000000, "bbox": [[0.825, 0.396, 0.861, 0.408], [0.139, 0.41, 0.861, 0.422], [0.139, 0.424, 0.383, 0.435]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 142, "edu_l1_label": "IOS"}, {"txt": "Recent applications have further demonstrated the versatility of LLE-based methods.Tao et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6100000, "bbox": [[0.163, 0.438, 0.83, 0.449]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 143, "edu_l1_label": "IOS"}, {"txt": "[15] proposed a privacy-preserving outsourcing scheme for face recognition, utilizing LLE with sparse orthogonal matrices to reduce computational complexity.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6100000, "bbox": [[0.83, 0.438, 0.861, 0.449], [0.139, 0.452, 0.861, 0.463], [0.139, 0.466, 0.435, 0.477]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 144, "edu_l1_label": "IOS"}, {"txt": "While the method enhances efciency through batch outsourcing and result verifcation, it faces limitations related to communication latency and scalability.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6100000, "bbox": [[0.435, 0.466, 0.861, 0.477], [0.139, 0.479, 0.748, 0.491]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 145, "edu_l1_label": "IOS"}, {"txt": "Wang et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6100000, "bbox": [[0.748, 0.479, 0.832, 0.491]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 146, "edu_l1_label": "IOS"}, {"txt": "[20] focused on improving graph structure learning with a non-negative orthogonal constraint.Their method enhances clustering accuracy by minimizing intra-class distances and optimizing inter-class topology through category anchors, though challenges related to scalability and initialization sensitivity remain.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6100000, "bbox": [[0.832, 0.479, 0.861, 0.491], [0.139, 0.493, 0.861, 0.505], [0.139, 0.507, 0.861, 0.518], [0.139, 0.521, 0.759, 0.532]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 147, "edu_l1_label": "IOS"}, {"txt": "The integration of metric learning into manifold learning ofers a signifcant advantage by allowing the distance function to adapt dynamically to the local geometry of the data.This fexibility enables better preservation of neighborhood relationships and improves embedding quality.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6200000, "bbox": [[0.163, 0.535, 0.861, 0.546], [0.139, 0.549, 0.861, 0.56], [0.139, 0.563, 0.643, 0.574]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 148, "edu_l1_label": "IOS"}, {"txt": "Our proposed method builds on this foundation by introducing a data-driven metric that adjusts dynamically to the structure of the data, leading to more accurate and robust low-dimensional representations.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6200000, "bbox": [[0.643, 0.563, 0.861, 0.574], [0.139, 0.576, 0.861, 0.588], [0.139, 0.59, 0.597, 0.602]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 149, "edu_l1_label": "IOS"}, {"txt": "3The proposed method", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6300000, "bbox": [[0.139, 0.627, 0.433, 0.645]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 7, "global_sentence_id": 150, "edu_l1_label": "BOS"}, {"txt": "In the realm of high-dimensional data analysis, traditional metrics like Euclidean distance often fall short, particularly when grappling with complex, non-linear datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6400000, "bbox": [[0.139, 0.66, 0.861, 0.671], [0.139, 0.674, 0.551, 0.685]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 151, "edu_l1_label": "IOS"}, {"txt": "The inherent assumption of uniform scaling in static metrics can distort neighborhood relationships, leading to misrepresentation in proximity calculations.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6400000, "bbox": [[0.551, 0.674, 0.861, 0.685], [0.139, 0.688, 0.861, 0.699]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 152, "edu_l1_label": "IOS"}, {"txt": "To overcome these challenges, we introduce a data-dependent metric that adapts to the intricate local geometry of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6400000, "bbox": [[0.139, 0.702, 0.861, 0.713], [0.139, 0.715, 0.218, 0.727]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 153, "edu_l1_label": "IOS"}, {"txt": "By learning a positive semi-defnite matrix M from the data, our approach not only enhances the fdelity of distance measurements but also preserves essential neighborhood structures, paving the way for more meaningful insights in manifold learning.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6400000, "bbox": [[0.218, 0.715, 0.861, 0.727], [0.139, 0.729, 0.861, 0.741], [0.139, 0.743, 0.446, 0.754]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 154, "edu_l1_label": "IOS"}, {"txt": "As illustrated in Fig.2, the process begins by initializing a metric matrix M that adapts to the unique structure of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6500000, "bbox": [[0.163, 0.757, 0.861, 0.768], [0.139, 0.771, 0.281, 0.782]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 155, "edu_l1_label": "IOS"}, {"txt": "This matrix is learned iteratively, starting with an initial guess that is refned through optimization.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6500000, "bbox": [[0.281, 0.771, 0.861, 0.782], [0.139, 0.785, 0.226, 0.796]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 156, "edu_l1_label": "IOS"}, {"txt": "The method involves fnding the K-nearest neighbors in the high-dimensional input space, guided by the learned metric M, rather than static distances.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6500000, "bbox": [[0.226, 0.785, 0.861, 0.796], [0.139, 0.799, 0.49, 0.81]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 157, "edu_l1_label": "IOS"}, {"txt": "This step helps to preserve essential topological features by capturing neighborhood relationships that refect the intrinsic geometry of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6500000, "bbox": [[0.49, 0.799, 0.861, 0.81], [0.139, 0.812, 0.715, 0.824]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 158, "edu_l1_label": "IOS"}, {"txt": "After identifying neighbors, we proceed to the linear reconstruction phase, where each data point is expressed as a weighted linear combination of its neighbors.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6600000, "bbox": [[0.163, 0.826, 0.861, 0.838], [0.139, 0.84, 0.516, 0.851]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 159, "edu_l1_label": "IOS"}, {"txt": "These weights are iteratively optimized to minimize reconstruction error while maintaining the local topology.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6600000, "bbox": [[0.516, 0.84, 0.861, 0.851], [0.139, 0.854, 0.521, 0.865]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 160, "edu_l1_label": "IOS"}, {"txt": "In parallel, the metric M is updated based on the reconstruction error, allowing it to better represent the data’s manifold structure in subsequent iterations.This alternating optimization loop between linear reconstruction and metric refnement ensures that both the neighborhood preservation and the accuracy of distance measurements are continually improved.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6600000, "bbox": [[0.521, 0.854, 0.861, 0.865], [0.139, 0.868, 0.861, 0.879], [0.139, 0.882, 0.861, 0.893], [0.139, 0.895, 0.777, 0.907]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 161, "edu_l1_label": "IOS"}, {"txt": "Finally, the optimized metric and reconstruction weights are used to generate the low-dimensional em-", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6700000, "bbox": [[0.163, 0.909, 0.861, 0.921]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 162, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6800000, "bbox": [[0.494, 0.957, 0.507, 0.968]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 163, "edu_l1_label": "EDU_O"}, {"txt": "bedding, preserving the manifold’s structure as faithfully as possible.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6800000, "bbox": [[0.139, 0.076, 0.598, 0.087]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 164, "edu_l1_label": "IOS"}, {"txt": "This adaptive approach, as detailed inFig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6800000, "bbox": [[0.598, 0.076, 0.861, 0.087], [0.139, 0.09, 0.165, 0.101]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 165, "edu_l1_label": "IOS"}, {"txt": "2, not only enhances the fdelity of distance measurements but also aligns them with the underlying data distribution, enabling more meaningful insights in manifold learning.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6800000, "bbox": [[0.165, 0.09, 0.861, 0.101], [0.139, 0.103, 0.592, 0.115]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 166, "edu_l1_label": "IOS"}, {"txt": "3.1Notations and defnitions", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 6900000, "bbox": [[0.139, 0.136, 0.432, 0.151]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 8, "global_sentence_id": 167, "edu_l1_label": "IOS"}, {"txt": "In this section, we introduce the key notations and defnitions that will be used throughout the paper.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.139, 0.161, 0.83, 0.172]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 168, "edu_l1_label": "IOS"}, {"txt": "Let X = {x1, x2, .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.83, 0.161, 0.861, 0.172], [0.139, 0.174, 0.229, 0.186]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 169, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.229, 0.174, 0.236, 0.186]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 170, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.236, 0.174, 0.242, 0.186]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 171, "edu_l1_label": "IOS"}, {"txt": ", xn} ⊂RD represent a set of n data points in a D-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.242, 0.174, 0.693, 0.186]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 172, "edu_l1_label": "IOS"}, {"txt": "Each data point xi ∈RDcorresponds to a high-dimensional vector.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.693, 0.174, 0.86, 0.18], [0.139, 0.188, 0.41, 0.2]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 173, "edu_l1_label": "IOS"}, {"txt": "In the context of LLE, let K denote the number of nearest neighbors considered for each data point.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.41, 0.188, 0.861, 0.2], [0.139, 0.202, 0.342, 0.213]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 174, "edu_l1_label": "IOS"}, {"txt": "For a given point xi, its set of KNN is denoted as N(xi).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7000000, "bbox": [[0.342, 0.202, 0.727, 0.213]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 175, "edu_l1_label": "IOS"}, {"txt": "The reconstruction process in LLE assigns weights to neighbors, where W ∈Rn×k is the matrix of re-construction weights.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7100000, "bbox": [[0.163, 0.216, 0.861, 0.227], [0.139, 0.23, 0.279, 0.241]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 176, "edu_l1_label": "IOS"}, {"txt": "Each entry wij represents the weight assigned to the j-th neighbor of the point xi.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7100000, "bbox": [[0.279, 0.23, 0.841, 0.241]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 177, "edu_l1_label": "IOS"}, {"txt": "A positive semi-defnite matrix M is introduced to represent the learned distance metric.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7100000, "bbox": [[0.841, 0.23, 0.861, 0.241], [0.139, 0.244, 0.715, 0.255]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 178, "edu_l1_label": "IOS"}, {"txt": "Using M, the Mahalanobis distance between two points xi and xj is defned as dM(xi, xj) =p(xi −xj)T M(xi −xj).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7100000, "bbox": [[0.715, 0.244, 0.861, 0.255], [0.139, 0.257, 0.79, 0.269]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 179, "edu_l1_label": "IOS"}, {"txt": "In the dimensionality reduction process, each point xi is mapped to a low-dimensional representation yi ∈Rd, where d is the dimension of the embedding space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7200000, "bbox": [[0.163, 0.271, 0.861, 0.283], [0.139, 0.285, 0.541, 0.296]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 180, "edu_l1_label": "IOS"}, {"txt": "The quality of the reconstruction in the high-dimensional space is quantifed using the reconstruction error, denoted as E(W).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7200000, "bbox": [[0.541, 0.285, 0.861, 0.296], [0.139, 0.299, 0.674, 0.31]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 181, "edu_l1_label": "IOS"}, {"txt": "This error is given by", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7200000, "bbox": [[0.674, 0.299, 0.823, 0.31]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 182, "edu_l1_label": "IOS"}, {"txt": "E ( \\mathbf { W } ) = \\sum _ { i = 1 } ^ { n } \\left\\| \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } w _ { i j } \\mathbf { x } _ { j } \\right\\| _ { \\mathbf { M } } ^ { 2 } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7300000, "bbox": [[0.376, 0.318, 0.618, 0.363]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 183, "edu_l1_label": "IOS"}, {"txt": "where ∥· ∥M is the norm induced by the learned metric M. For each point xi, the reconstruction error vectoris defned as ri = xi −Pj∈N (i) wijxj, capturing the deviation of xi from its weighted reconstruction based onits neighbors.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7400000, "bbox": [[0.139, 0.377, 0.861, 0.389], [0.139, 0.391, 0.861, 0.402], [0.139, 0.405, 0.227, 0.416]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 184, "edu_l1_label": "IOS"}, {"txt": "These notations and defnitions form the basis for the subsequent discussions on the proposed modifcations and optimization processes in LLE.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7400000, "bbox": [[0.227, 0.405, 0.861, 0.416], [0.139, 0.419, 0.465, 0.43]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 185, "edu_l1_label": "IOS"}, {"txt": "3.2Problem Formulation", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7500000, "bbox": [[0.139, 0.451, 0.395, 0.466]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 9, "global_sentence_id": 186, "edu_l1_label": "IOS"}, {"txt": "In the classical LLE algorithm proposed by Roweis and Saul [14], the goal is to reduce the dimensionality of a dataset [7] while preserving the local neighborhood structure of the data through three key steps.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.139, 0.476, 0.861, 0.487], [0.139, 0.49, 0.812, 0.501]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 187, "edu_l1_label": "IOS"}, {"txt": "Given a set of points {x1, x2, .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.812, 0.49, 0.861, 0.501], [0.139, 0.504, 0.3, 0.515]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 188, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.3, 0.504, 0.307, 0.515]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 189, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.307, 0.504, 0.314, 0.515]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 190, "edu_l1_label": "IOS"}, {"txt": ", xn} ∈RD, the algorithm frst identifes the KNN of each data point [2].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.314, 0.504, 0.817, 0.515]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 191, "edu_l1_label": "IOS"}, {"txt": "Oncethe neighbors are determined, LLE computes a set of reconstruction weights that express each point xi as a linear combination of its neighbors in the original high-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.817, 0.504, 0.861, 0.515], [0.139, 0.517, 0.861, 0.529], [0.139, 0.531, 0.629, 0.543]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 192, "edu_l1_label": "IOS"}, {"txt": "These weights are then utilized to construct a low-dimensional embedding {y1, y2, .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.629, 0.531, 0.861, 0.543], [0.139, 0.545, 0.465, 0.556]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 193, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.465, 0.545, 0.472, 0.556]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 194, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.472, 0.545, 0.479, 0.556]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 195, "edu_l1_label": "IOS"}, {"txt": ", yN} ∈Rd (with d ≪D), ensuring that each point yi issimilarly represented as a weighted combination of its neighbors, thereby preserving the local structure of the data in the lower-dimensional representation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.479, 0.545, 0.861, 0.556], [0.139, 0.559, 0.861, 0.57], [0.139, 0.573, 0.437, 0.584]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 196, "edu_l1_label": "IOS"}, {"txt": "while the The standard LLE algorithm determines neighboring points using a fxed distance metric, typically the Euclidean distance:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7600000, "bbox": [[0.437, 0.573, 0.861, 0.584], [0.139, 0.587, 0.597, 0.598]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 197, "edu_l1_label": "IOS"}, {"txt": "d(x1, xj) = ||x1 + xj||2.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7700000, "bbox": [[0.427, 0.61, 0.573, 0.625]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 13, "global_sentence_id": 198, "edu_l1_label": "IOS"}, {"txt": "This fxed notion of distance may fail to accurately capture the true local geometry of the data, particularly in cases where the manifold is curved or the data is anisotropically distributed.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7800000, "bbox": [[0.139, 0.636, 0.861, 0.647], [0.139, 0.65, 0.659, 0.661]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 199, "edu_l1_label": "IOS"}, {"txt": "To address this limitation, we propose a controllable LLE framework in which the distance between points is governed by a learned metric.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7800000, "bbox": [[0.659, 0.65, 0.861, 0.661], [0.139, 0.663, 0.861, 0.675]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 200, "edu_l1_label": "IOS"}, {"txt": "By utilizing a Mahalanobis distance metric in both the K-nearest neighbors search and the linear reconstruction steps, our approach enhances the algorithm’s capacity to adaptively model the local structure of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7800000, "bbox": [[0.139, 0.677, 0.861, 0.689], [0.139, 0.691, 0.825, 0.702]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 201, "edu_l1_label": "IOS"}, {"txt": "Thismethodology facilitates a more precise representation of the underlying manifold, as will be discussed further.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7800000, "bbox": [[0.825, 0.691, 0.861, 0.702], [0.139, 0.705, 0.855, 0.716]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 202, "edu_l1_label": "IOS"}, {"txt": "3.3Initialization", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 7900000, "bbox": [[0.139, 0.737, 0.311, 0.752]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 11, "global_sentence_id": 203, "edu_l1_label": "IOS"}, {"txt": "As illustrated int Fig.3, this phase is a critical preparatory step it involves two sub-steps:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8000000, "bbox": [[0.139, 0.762, 0.727, 0.773]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 204, "edu_l1_label": "IOS"}, {"txt": "3.3.1Find Nearest Neighbors", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8100000, "bbox": [[0.139, 0.794, 0.393, 0.806]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 10, "global_sentence_id": 205, "edu_l1_label": "IOS"}, {"txt": "In this step, the nearest neighbors of each data point in the high-dimensional space are identifed, forming the local neighborhood that is preserved during the dimensionality reduction process.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8200000, "bbox": [[0.139, 0.817, 0.861, 0.828], [0.139, 0.831, 0.707, 0.842]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 206, "edu_l1_label": "IOS"}, {"txt": "To fnd the neighbors, a distance metric is employed to measure proximity between points, with the Euclidean distance being the most commonly used, although alternative metrics can be applied depending on the application.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8200000, "bbox": [[0.707, 0.831, 0.861, 0.842], [0.139, 0.844, 0.861, 0.856], [0.139, 0.858, 0.766, 0.87]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 207, "edu_l1_label": "IOS"}, {"txt": "For each data point xi, the distances to all other points are computed, and the K closest points are selected as its neighbors.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8200000, "bbox": [[0.766, 0.858, 0.861, 0.87], [0.139, 0.872, 0.861, 0.883]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 208, "edu_l1_label": "IOS"}, {"txt": "These neighborhoods are then stored for use in subsequent steps, such as computing reconstruction weights and embeddings, ensuring the local structure of the data is efectively captured and the manifold’s geometryis preserved in the lower-dimensional representation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8200000, "bbox": [[0.139, 0.886, 0.861, 0.897], [0.139, 0.9, 0.861, 0.911], [0.139, 0.914, 0.485, 0.925]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 209, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8300000, "bbox": [[0.494, 0.956, 0.507, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 210, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8300000, "bbox": [[0.311, 0.068, 0.684, 0.336]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 1, "global_sentence_id": 211, "edu_l1_label": "EDU_O"}, {"txt": "Figure 3: Initialization", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8400000, "bbox": [[0.416, 0.343, 0.584, 0.359]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 8, "global_sentence_id": 212, "edu_l1_label": "EDU_O"}, {"txt": "3.3.2Initialize Metric", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8500000, "bbox": [[0.139, 0.387, 0.329, 0.4]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 6, "global_sentence_id": 213, "edu_l1_label": "IOS"}, {"txt": "In the process of LLE with metric learning, the initial metric M can be set in several ways.The most straightforward approach is the Euclidean initialization, where the metric M is initialized to the identity matrix I, corresponding to the standard Euclidean distance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8600000, "bbox": [[0.139, 0.41, 0.861, 0.422], [0.139, 0.424, 0.861, 0.435], [0.139, 0.438, 0.535, 0.449]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 214, "edu_l1_label": "IOS"}, {"txt": "Mathematically, this is expressed as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8600000, "bbox": [[0.535, 0.438, 0.782, 0.449]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 215, "edu_l1_label": "IOS"}, {"txt": "M = I", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8700000, "bbox": [[0.478, 0.461, 0.522, 0.473]]}]}, "tags": ["title"], "label": "formula", "web_segment_id": 12, "global_sentence_id": 216, "edu_l1_label": "IOS"}, {"txt": "In this case, the distance between points xi and xj is equivalent to the Euclidean distance:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8800000, "bbox": [[0.139, 0.485, 0.737, 0.496]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 10, "global_sentence_id": 217, "edu_l1_label": "IOS"}, {"txt": "dM(X1, xj) = ||x1 + xj|2", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 8900000, "bbox": [[0.42, 0.505, 0.578, 0.522]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 218, "edu_l1_label": "IOS"}, {"txt": "This initialization assumes no initial distortion or learning in the distance metric.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9000000, "bbox": [[0.139, 0.532, 0.679, 0.543]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 219, "edu_l1_label": "IOS"}, {"txt": "An alternative approach is random initialization, where M is initialized as M = LL⊤, with L being a randomly initialized matrix.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9000000, "bbox": [[0.679, 0.532, 0.861, 0.543], [0.139, 0.546, 0.824, 0.557]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 220, "edu_l1_label": "IOS"}, {"txt": "Thisensures that M is PSD, a desirable property for a valid distance metric.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9000000, "bbox": [[0.824, 0.546, 0.861, 0.557], [0.139, 0.56, 0.63, 0.571]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 221, "edu_l1_label": "IOS"}, {"txt": "Specifcally, each element of L is typically drawn from a normal distribution:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9000000, "bbox": [[0.63, 0.56, 0.861, 0.571], [0.139, 0.573, 0.427, 0.585]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 222, "edu_l1_label": "IOS"}, {"txt": "Lij~N(0,02)", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9100000, "bbox": [[0.451, 0.595, 0.549, 0.61]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 13, "global_sentence_id": 223, "edu_l1_label": "IOS"}, {"txt": "where N(0, σ2) is a normal distribution with mean 0 and variance σ2.This guarantees that M remainssymmetric and PSD:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9200000, "bbox": [[0.136, 0.615, 0.865, 0.66]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 11, "global_sentence_id": 224, "edu_l1_label": "IOS"}, {"txt": " M = LL⊤", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9200000, "bbox": [[0.136, 0.615, 0.865, 0.66]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 11, "global_sentence_id": 225, "edu_l1_label": "IOS"}, {"txt": "3.4Optimization Problem", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9300000, "bbox": [[0.139, 0.685, 0.404, 0.7]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 5, "global_sentence_id": 226, "edu_l1_label": "IOS"}, {"txt": "As illustrated int Fig.4 optimization process in proposed approach consists of two key components:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9400000, "bbox": [[0.139, 0.71, 0.807, 0.721]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 227, "edu_l1_label": "IOS"}, {"txt": " Reconstruction, which reconstructs each data point from its neighbors, and metric learning, which seeks to identify an optimal metric based on the reconstruction error.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9400000, "bbox": [[0.807, 0.71, 0.861, 0.721], [0.139, 0.723, 0.861, 0.735], [0.139, 0.737, 0.489, 0.749]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 228, "edu_l1_label": "IOS"}, {"txt": "These two steps are iteratively repeated within a loop, and after a predefned number of epochs, the optimization process is completed.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9400000, "bbox": [[0.489, 0.737, 0.861, 0.749], [0.139, 0.751, 0.667, 0.762]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 229, "edu_l1_label": "IOS"}, {"txt": "3.4.1Reconstruction", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9500000, "bbox": [[0.139, 0.783, 0.32, 0.795]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 7, "global_sentence_id": 230, "edu_l1_label": "IOS"}, {"txt": "Let xij ∈Rd denote the j-th neighbor of xi, and let the matrix Rd×k ∋Xi :", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9600000, "bbox": [[0.139, 0.807, 0.665, 0.819]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 231, "edu_l1_label": "IOS"}, {"txt": "= [xi1, .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9600000, "bbox": [[0.665, 0.807, 0.718, 0.819]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 232, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9600000, "bbox": [[0.718, 0.807, 0.725, 0.819]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 233, "edu_l1_label": "IOS"}, {"txt": ".", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9600000, "bbox": [[0.725, 0.807, 0.732, 0.819]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 234, "edu_l1_label": "IOS"}, {"txt": ", xik] include the kneighbors of xi.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9600000, "bbox": [[0.732, 0.807, 0.861, 0.819], [0.139, 0.821, 0.242, 0.833]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 235, "edu_l1_label": "IOS"}, {"txt": "\\operatorname* { m i n } _ { \\tilde { \\mathbf { W } } } E ( \\mathbf { W } ) = \\sum _ { i = 1 } ^ { N } \\left\\| \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { x } _ { i j } \\right\\| _ { \\mathbf { M } } ^ { 2 } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9700000, "bbox": [[0.358, 0.831, 0.638, 0.878]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 15, "global_sentence_id": 236, "edu_l1_label": "IOS"}, {"txt": "after optimization, we would have optimized W˜ as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9800000, "bbox": [[0.139, 0.884, 0.476, 0.895]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 237, "edu_l1_label": "IOS"}, {"txt": "\\mathbf { \\tilde { w } } _ { i } = \\frac { \\lambda _ { i } } { 2 } \\mathbf { G } _ { i } ^ { - 1 } \\mathbf { 1 } = \\frac { \\mathbf { G } _ { i } ^ { - 1 } \\mathbf { 1 } } { \\mathbf { 1 } ^ { \\top } \\mathbf { G } _ { i } ^ { - 1 } \\mathbf { 1 } }", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 9900000, "bbox": [[0.406, 0.901, 0.588, 0.935]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 16, "global_sentence_id": 238, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10000000, "bbox": [[0.493, 0.955, 0.507, 0.968]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 239, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10000000, "bbox": [[0.131, 0.067, 0.864, 0.333]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 0, "global_sentence_id": 240, "edu_l1_label": "EDU_O"}, {"txt": "Figure 4: Optimization Problem.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10100000, "bbox": [[0.38, 0.341, 0.62, 0.358]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 2, "global_sentence_id": 241, "edu_l1_label": "EDU_O"}, {"txt": "where:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10200000, "bbox": [[0.139, 0.387, 0.182, 0.398]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 11, "global_sentence_id": 242, "edu_l1_label": "IOS"}, {"txt": "\\begin{aligned} { \\mathbb { R } ^ { k \\times k } \\ni \\mathbf { G } _ { i } : = \\left( \\mathbf { x } _ { i } \\mathbf { 1 } ^ { \\top } - \\mathbf { X } _ { i } \\right) ^ { \\top } \\mathbf { M } ( \\mathbf { x } _ { i } \\mathbf { 1 } ^ { \\top } - \\mathbf { X } _ { i } ) } \\\\ { \\mathbf { x } _ { i } = \\mathbf { x } _ { i } \\mathbf { 1 } ^ { \\top } \\mathbf { \\tilde { w } } _ { i } . } \\\\ \\end{aligned}", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10300000, "bbox": [[0.345, 0.39, 0.654, 0.427]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 17, "global_sentence_id": 243, "edu_l1_label": "IOS"}, {"txt": "3.4.2 Metric Optimization", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10400000, "bbox": [[0.139, 0.451, 0.177, 0.464], [0.196, 0.451, 0.365, 0.464]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 8, "global_sentence_id": 244, "edu_l1_label": "IOS"}, {"txt": "We defne the objective function for optimizing the metric matrix M with fxed weights W:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10500000, "bbox": [[0.139, 0.474, 0.745, 0.486]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 245, "edu_l1_label": "IOS"}, {"txt": "\\operatorname* { m i n } _ { \\mathbf { M } } E ( \\mathbf { M } ) = \\sum _ { i = 1 } ^ { N } \\left\\| \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { x } _ { j } \\right\\| _ { \\mathbf { M } } ^ { 2 } .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10600000, "bbox": [[0.362, 0.493, 0.635, 0.54]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 246, "edu_l1_label": "IOS"}, {"txt": "The resulting closed-form gradient descent update for M is given by:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10700000, "bbox": [[0.139, 0.551, 0.594, 0.563]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 247, "edu_l1_label": "IOS"}, {"txt": "\\mathbf { M } ^ { ( t + 1 ) } = \\mathbf { M } ^ { ( t ) } - \\eta \\sum _ { i = 1 } ^ { N } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10800000, "bbox": [[0.402, 0.57, 0.592, 0.609]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 15, "global_sentence_id": 248, "edu_l1_label": "IOS"}, {"txt": "where ri represents the residual.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10900000, "bbox": [[0.139, 0.621, 0.352, 0.632]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 249, "edu_l1_label": "IOS"}, {"txt": "(For derivation, refer to Appendix A.1).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 10900000, "bbox": [[0.352, 0.621, 0.622, 0.632]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 250, "edu_l1_label": "IOS"}, {"txt": "3.4.3 Linear Embedding", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11000000, "bbox": [[0.139, 0.653, 0.177, 0.665], [0.196, 0.653, 0.347, 0.665]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 1, "global_sentence_id": 251, "edu_l1_label": "IOS"}, {"txt": "Our objective for embedding data into a lower-dimensional space with fxed weights W is:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11100000, "bbox": [[0.139, 0.676, 0.735, 0.687]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 252, "edu_l1_label": "IOS"}, {"txt": "\\operatorname* { m i n } _ { \\mathbf { Y } } \\sum _ { i = 1 } ^ { N } \\left\\| \\mathbf { y } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { y } _ { j } \\right\\| _ { 2 } ^ { 2 } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11200000, "bbox": [[0.394, 0.695, 0.595, 0.742]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 16, "global_sentence_id": 253, "edu_l1_label": "IOS"}, {"txt": "n n The pronbPlemi= s1implifes to an Peigie=n1value decomposition:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11300000, "bbox": [[0.24, 0.752, 0.247, 0.76], [0.372, 0.752, 0.38, 0.76], [0.163, 0.769, 0.528, 0.78]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 254, "edu_l1_label": "IOS"}, {"txt": "MWY = XY,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11400000, "bbox": [[0.454, 0.792, 0.545, 0.805]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 13, "global_sentence_id": 255, "edu_l1_label": "IOS"}, {"txt": "where MW = (I −W)T (I −W).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11500000, "bbox": [[0.139, 0.818, 0.356, 0.829]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 256, "edu_l1_label": "IOS"}, {"txt": "(For detailed derivation, refer to Appendix A.2).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11500000, "bbox": [[0.356, 0.818, 0.683, 0.829]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 257, "edu_l1_label": "IOS"}, {"txt": "The solution Y is constructed using the eigenvectors corresponding to the d smallest non-zero eigenvalues (excluding the smallest eigenvalue, which is zero due to the translation invariance of MW ):", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11600000, "bbox": [[0.163, 0.831, 0.861, 0.843], [0.139, 0.845, 0.738, 0.857]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 258, "edu_l1_label": "IOS"}, {"txt": "where vi is the i-th eigenvector of MW .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11700000, "bbox": [[0.139, 0.897, 0.398, 0.908]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 10, "global_sentence_id": 259, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11800000, "bbox": [[0.494, 0.956, 0.507, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 260, "edu_l1_label": "EDU_O"}, {"txt": "3.5Dimension-Reduced Data", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 11900000, "bbox": [[0.139, 0.073, 0.435, 0.088]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 9, "global_sentence_id": 261, "edu_l1_label": "IOS"}, {"txt": "The reduced-dimensional representation of the data is:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12000000, "bbox": [[0.139, 0.098, 0.499, 0.109]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 262, "edu_l1_label": "IOS"}, {"txt": "where yi ∈Rd is the i-th row of Y .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12100000, "bbox": [[0.139, 0.147, 0.369, 0.158]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 13, "global_sentence_id": 263, "edu_l1_label": "IOS"}, {"txt": "3.6Theoretical Guarantee:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12200000, "bbox": [[0.139, 0.179, 0.411, 0.194]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 11, "global_sentence_id": 264, "edu_l1_label": "IOS"}, {"txt": " Ensuring Positive Semi-Defniteness of M", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12200000, "bbox": [[0.411, 0.179, 0.82, 0.194]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 11, "global_sentence_id": 265, "edu_l1_label": "IOS"}, {"txt": "In this subsection, we provide a theoretical guarantee that the matrix M, defned as M = LT L, remains PSD throughout the optimization process.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12300000, "bbox": [[0.139, 0.207, 0.861, 0.218], [0.139, 0.22, 0.381, 0.232]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 266, "edu_l1_label": "IOS"}, {"txt": "To achieve this, we propose an alternative approach: instead of updating M directly, we update L, which guarantees that M is always PSD.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12300000, "bbox": [[0.381, 0.22, 0.861, 0.232], [0.139, 0.234, 0.58, 0.246]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 267, "edu_l1_label": "IOS"}, {"txt": "To guarantee that M remains PSD, we propose updating L instead of M. This ensures that the product LT L remains PSD throughout the optimization.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12400000, "bbox": [[0.163, 0.248, 0.861, 0.259], [0.139, 0.262, 0.456, 0.273]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 268, "edu_l1_label": "IOS"}, {"txt": "The gradient of the objective function E(M) with respect to L is derived as follows:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12400000, "bbox": [[0.456, 0.262, 0.861, 0.273], [0.139, 0.276, 0.289, 0.287]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 269, "edu_l1_label": "IOS"}, {"txt": "E ( \\mathbf { M } ) = \\sum _ { i = 1 } ^ { N } \\mathbf { r } _ { i } ^ { T } \\mathbf { L } ^ { T } \\mathbf { L } \\mathbf { r } _ { i } = \\sum _ { i = 1 } ^ { N } ( \\mathbf { L } \\mathbf { r } _ { i } ) ^ { T } ( \\mathbf { L } \\mathbf { r } _ { i } )", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12500000, "bbox": [[0.362, 0.295, 0.635, 0.336]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 270, "edu_l1_label": "IOS"}, {"txt": "Taking the derivative of E(M) with respect to L, we get:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12600000, "bbox": [[0.163, 0.341, 0.541, 0.352]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 10, "global_sentence_id": 271, "edu_l1_label": "IOS"}, {"txt": "\\frac { \\partial E ( \\mathbf { M } ) } { \\partial \\mathbf { L } } = 2 \\sum _ { i = 1 } ^ { N } \\mathbf { L } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T }", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12700000, "bbox": [[0.42, 0.361, 0.573, 0.399]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 15, "global_sentence_id": 272, "edu_l1_label": "IOS"}, {"txt": "This gradient expression allows us to update L in the optimization process.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12800000, "bbox": [[0.163, 0.405, 0.656, 0.416]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 273, "edu_l1_label": "IOS"}, {"txt": "We update the matrix L using the following gradient descent update rule:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12800000, "bbox": [[0.656, 0.405, 0.861, 0.416], [0.139, 0.419, 0.421, 0.43]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 274, "edu_l1_label": "IOS"}, {"txt": "\\mathbf { L } ^ { ( t + 1 ) } = \\mathbf { L } ^ { ( t ) } - 2 \\eta \\sum _ { i = 1 } ^ { N } \\mathbf { L } ^ { ( t ) } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T }", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 12900000, "bbox": [[0.394, 0.437, 0.604, 0.476]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 16, "global_sentence_id": 275, "edu_l1_label": "IOS"}, {"txt": "where η is the learning rate.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13000000, "bbox": [[0.163, 0.485, 0.352, 0.496]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 276, "edu_l1_label": "IOS"}, {"txt": "By updating L in this manner, we guarantee that M = LT L remains PSDat every iteration.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13000000, "bbox": [[0.352, 0.485, 0.861, 0.496], [0.139, 0.499, 0.26, 0.51]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 277, "edu_l1_label": "IOS"}, {"txt": "This is because M is the product of L and its transpose, and such a product is always positive semi-defnite.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13000000, "bbox": [[0.26, 0.499, 0.861, 0.51], [0.139, 0.513, 0.281, 0.524]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 278, "edu_l1_label": "IOS"}, {"txt": "In some cases, to ensure that M does not become singular or degenerate (i.e., with zero or near-zero eigenvalues), we can introduce a regularization term.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13000000, "bbox": [[0.281, 0.513, 0.861, 0.524], [0.139, 0.526, 0.571, 0.538]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 279, "edu_l1_label": "IOS"}, {"txt": "The updated rule for L with regularization becomes:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13000000, "bbox": [[0.571, 0.526, 0.861, 0.538], [0.139, 0.54, 0.198, 0.552]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 280, "edu_l1_label": "IOS"}, {"txt": "\\mathbf { L } ^ { ( t + 1 ) } = \\mathbf { L } ^ { ( t ) } - 2 \\eta \\sum _ { i = 1 } ^ { N } \\mathbf { L } ^ { ( t ) } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T } + \\lambda \\mathbf { L } ^ { ( t ) }", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13100000, "bbox": [[0.364, 0.559, 0.627, 0.598]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 17, "global_sentence_id": 281, "edu_l1_label": "IOS"}, {"txt": "where λ > 0 is a small positive constant that ensures that L remains well-conditioned.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13200000, "bbox": [[0.163, 0.604, 0.73, 0.616]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 282, "edu_l1_label": "IOS"}, {"txt": "This regularizationterm ensures that the eigenvalues of M do not become too small, maintaining the stability of the optimization process.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13200000, "bbox": [[0.73, 0.604, 0.861, 0.616], [0.139, 0.618, 0.861, 0.63], [0.139, 0.632, 0.191, 0.643]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 283, "edu_l1_label": "IOS"}, {"txt": "The fnal algorithm of the proposed method is presented in Algorithm 1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13200000, "bbox": [[0.191, 0.632, 0.677, 0.643]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 284, "edu_l1_label": "IOS"}, {"txt": "4Experimental results", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13300000, "bbox": [[0.139, 0.669, 0.418, 0.687]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 5, "global_sentence_id": 285, "edu_l1_label": "BOS"}, {"txt": "In this section, we present the evaluation metrics and datasets employed in our study.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13400000, "bbox": [[0.139, 0.702, 0.687, 0.713]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 286, "edu_l1_label": "IOS"}, {"txt": "Following this, we provide a comparative analysis of the results obtained from our proposed approach against those of the standard LLE and ISO LLE algorithms.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13400000, "bbox": [[0.687, 0.702, 0.861, 0.713], [0.139, 0.716, 0.861, 0.727], [0.139, 0.73, 0.307, 0.741]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 287, "edu_l1_label": "IOS"}, {"txt": "4.1Evaluation Metrics", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13500000, "bbox": [[0.139, 0.762, 0.372, 0.777]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 7, "global_sentence_id": 288, "edu_l1_label": "IOS"}, {"txt": "To assess the performance of the proposed Consistent LLE algorithm, we employ several evaluation metrics.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13600000, "bbox": [[0.139, 0.787, 0.861, 0.798]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 289, "edu_l1_label": "IOS"}, {"txt": "These metrics capture various aspects of the algorithm’s efectiveness, including the preservation of local andglobal data structures and the suitability of the learned low-dimensional representations for downstream tasks like clustering and classifcation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13600000, "bbox": [[0.139, 0.801, 0.861, 0.812], [0.139, 0.815, 0.861, 0.826], [0.139, 0.828, 0.352, 0.84]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 290, "edu_l1_label": "IOS"}, {"txt": "The following metrics are used in our evaluation:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13600000, "bbox": [[0.352, 0.828, 0.681, 0.84]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 291, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13700000, "bbox": [[0.494, 0.956, 0.507, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 292, "edu_l1_label": "EDU_O"}, {"txt": "header", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13700000, "bbox": [[0.138, 0.072, 0.293, 0.087]]}]}, "tags": ["header"], "label": "O", "web_segment_id": -1, "global_sentence_id": 293, "edu_l1_label": "EDU_O"}, {"txt": "2:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.111, 0.158, 0.121]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 19, "global_sentence_id": 294, "edu_l1_label": "IOS"}, {"txt": " Output: Lower-dimensional embedding of the data", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.111, 0.537, 0.121]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 19, "global_sentence_id": 295, "edu_l1_label": "IOS"}, {"txt": "3:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.126, 0.158, 0.136]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 11, "global_sentence_id": 296, "edu_l1_label": "EDU_O"}, {"txt": " Initialize the metric matrix M.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.126, 0.388, 0.136]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 11, "global_sentence_id": 297, "edu_l1_label": "EDU_O"}, {"txt": "4:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.141, 0.158, 0.151]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 12, "global_sentence_id": 298, "edu_l1_label": "EDU_O"}, {"txt": " Initialize M as the identity matrix.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.141, 0.417, 0.152]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 299, "edu_l1_label": "EDU_O"}, {"txt": "5:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.156, 0.158, 0.166]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 17, "global_sentence_id": 300, "edu_l1_label": "EDU_O"}, {"txt": " Find nneighbors nearest neighbors for each data point.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.156, 0.545, 0.167]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 17, "global_sentence_id": 301, "edu_l1_label": "EDU_O"}, {"txt": "6:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.171, 0.158, 0.181]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 22, "global_sentence_id": 302, "edu_l1_label": "EDU_O"}, {"txt": " for epoch = 1 to max epochs do", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.171, 0.401, 0.182]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 22, "global_sentence_id": 303, "edu_l1_label": "EDU_O"}, {"txt": "7:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.147, 0.183, 0.588, 0.196]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 15, "global_sentence_id": 304, "edu_l1_label": "IOS"}, {"txt": " Call CholeskyDecomposition to Decompose Metric M", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.147, 0.183, 0.588, 0.196]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 15, "global_sentence_id": 305, "edu_l1_label": "IOS"}, {"txt": "8:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.148, 0.201, 0.158, 0.211]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 21, "global_sentence_id": 306, "edu_l1_label": "IOS"}, {"txt": "Compute reconstruction weights using LinearReconstruction() with respect to the metric M.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.201, 0.861, 0.212], [0.167, 0.214, 0.189, 0.227]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 21, "global_sentence_id": 307, "edu_l1_label": "IOS"}, {"txt": "9:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.166, 0.227, 0.483, 0.241]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 18, "global_sentence_id": 308, "edu_l1_label": "EDU_O"}, {"txt": " if Optimization method = ’Adam’ then", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.166, 0.227, 0.483, 0.241]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 18, "global_sentence_id": 309, "edu_l1_label": "EDU_O"}, {"txt": "10:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.14, 0.243, 0.638, 0.256]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 20, "global_sentence_id": 310, "edu_l1_label": "EDU_O"}, {"txt": " Update L using the Adam optimizer via AdamOptimizer().", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.14, 0.243, 0.638, 0.256]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 20, "global_sentence_id": 311, "edu_l1_label": "EDU_O"}, {"txt": "else if Optimization method = ’SGD’ then", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.191, 0.26, 0.507, 0.272]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 23, "global_sentence_id": 312, "edu_l1_label": "EDU_O"}, {"txt": "12:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.14, 0.273, 0.736, 0.288]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 24, "global_sentence_id": 313, "edu_l1_label": "IOS"}, {"txt": " Update L using stochastic gradient descent (SGD) via SGDOptimizer().", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.14, 0.273, 0.736, 0.288]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 24, "global_sentence_id": 314, "edu_l1_label": "IOS"}, {"txt": "13:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.138, 0.289, 0.243, 0.302]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 25, "global_sentence_id": 315, "edu_l1_label": "EDU_O"}, {"txt": " end if", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.138, 0.289, 0.243, 0.302]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 25, "global_sentence_id": 316, "edu_l1_label": "EDU_O"}, {"txt": "14:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.141, 0.307, 0.158, 0.317]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 16, "global_sentence_id": 317, "edu_l1_label": "EDU_O"}, {"txt": " end for", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.307, 0.224, 0.318]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 16, "global_sentence_id": 318, "edu_l1_label": "EDU_O"}, {"txt": "15:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.141, 0.322, 0.158, 0.332]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 14, "global_sentence_id": 319, "edu_l1_label": "IOS"}, {"txt": " Perform linear embedding using LinearEmbedding().", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.322, 0.55, 0.333]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 14, "global_sentence_id": 320, "edu_l1_label": "IOS"}, {"txt": "16:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.141, 0.337, 0.158, 0.347]]}]}, "tags": ["text"], "label": "code", "web_segment_id": 13, "global_sentence_id": 321, "edu_l1_label": "IOS"}, {"txt": " Return: Lower-dimensional embedding of the data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13800000, "bbox": [[0.158, 0.337, 0.54, 0.348]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 13, "global_sentence_id": 322, "edu_l1_label": "IOS"}, {"txt": "4.1.1Neighborhood Preservation", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 13900000, "bbox": [[0.139, 0.381, 0.422, 0.393]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 10, "global_sentence_id": 323, "edu_l1_label": "IOS"}, {"txt": "To evaluate how well the Consistent LLE preserves local neighborhood structures, we employ two standard metrics:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14000000, "bbox": [[0.139, 0.404, 0.861, 0.415], [0.139, 0.418, 0.191, 0.429]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 324, "edu_l1_label": "IOS"}, {"txt": " Trustworthiness and Continuity, introduced by Venna and Kaski [19].", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14000000, "bbox": [[0.191, 0.418, 0.658, 0.429]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 325, "edu_l1_label": "IOS"}, {"txt": "• Trustworthiness quantifes how many of the original nearest neighbors in high-dimensional space areretained as neighbors in the low-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14100000, "bbox": [[0.164, 0.434, 0.861, 0.448], [0.18, 0.45, 0.518, 0.462]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 326, "edu_l1_label": "IOS"}, {"txt": "It is given by:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14100000, "bbox": [[0.518, 0.45, 0.615, 0.462]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 327, "edu_l1_label": "IOS"}, {"txt": "T ( k ) = 1 - \\frac { 2 } { n k ( 2 n - 3 k - 1 ) } \\sum _ { i = 1 } ^ { n } \\sum _ { j \\in U ( i , k ) } ( r ( i , j ) - k )", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14200000, "bbox": [[0.344, 0.47, 0.691, 0.509]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 29, "global_sentence_id": 328, "edu_l1_label": "IOS"}, {"txt": "where r(i, j) is the rank of point j in the neighborhood of point i in the original high-dimensional space, and U(i, k) is the set of points in the low-dimensional space that are not among the k-nearest neighbors of point i.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14300000, "bbox": [[0.18, 0.52, 0.861, 0.532], [0.18, 0.534, 0.861, 0.546], [0.18, 0.548, 0.245, 0.559]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 329, "edu_l1_label": "IOS"}, {"txt": "A higher trustworthiness score indicates better preservation of local neighborhoods.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14300000, "bbox": [[0.245, 0.548, 0.803, 0.559]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 330, "edu_l1_label": "IOS"}, {"txt": "• Continuity evaluates how well the low-dimensional neighbors are true neighbors in the original space.Continuity is computed as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14400000, "bbox": [[0.164, 0.565, 0.861, 0.578], [0.18, 0.581, 0.357, 0.592]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 331, "edu_l1_label": "IOS"}, {"txt": "2", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14500000, "bbox": [[0.479, 0.605, 0.49, 0.616]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 27, "global_sentence_id": 332, "edu_l1_label": "EDU_O"}, {"txt": "√", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14500000, "bbox": [[0.544, 0.615, 0.552, 0.621]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 26, "global_sentence_id": 333, "edu_l1_label": "EDU_O"}, {"txt": "where s(i, j) is the rank of point j in the low-dimensional space relative to point i, and V (i, k) represents the neighbors in the original space that are missing in the low-dimensional embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14600000, "bbox": [[0.18, 0.651, 0.861, 0.662], [0.18, 0.665, 0.741, 0.676]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 334, "edu_l1_label": "IOS"}, {"txt": "Higher continuity indicates that the neighborhood structure is preserved well in both directions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14600000, "bbox": [[0.741, 0.665, 0.861, 0.676], [0.18, 0.679, 0.694, 0.69]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 335, "edu_l1_label": "IOS"}, {"txt": "4.1.2Silhouette Score", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14700000, "bbox": [[0.139, 0.71, 0.329, 0.723]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 9, "global_sentence_id": 336, "edu_l1_label": "IOS"}, {"txt": "The silhouette score [13] is used to evaluate the quality of clustering in the low-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14800000, "bbox": [[0.139, 0.733, 0.784, 0.745]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 337, "edu_l1_label": "IOS"}, {"txt": "It provides a measure of how similar an object is to its own cluster compared to other clusters.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14800000, "bbox": [[0.784, 0.733, 0.861, 0.745], [0.139, 0.747, 0.696, 0.759]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 338, "edu_l1_label": "IOS"}, {"txt": "The silhouette score for each data point is given by:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14800000, "bbox": [[0.696, 0.747, 0.861, 0.759], [0.139, 0.761, 0.321, 0.772]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 339, "edu_l1_label": "IOS"}, {"txt": "where a(i) is the average intra-cluster distance (the average distance between the point i and other points in its cluster), and b(i) is the average nearest-cluster distance (the average distance between point i and points in the nearest cluster).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14900000, "bbox": [[0.139, 0.806, 0.861, 0.817], [0.139, 0.819, 0.861, 0.831], [0.139, 0.833, 0.289, 0.845]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 340, "edu_l1_label": "IOS"}, {"txt": "The silhouette score ranges from -1 to 1, with higher values indicating better-defned clusters.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14900000, "bbox": [[0.289, 0.833, 0.861, 0.845], [0.139, 0.847, 0.193, 0.858]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 341, "edu_l1_label": "IOS"}, {"txt": "We compute the overall silhouette score as the mean score across all data points.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 14900000, "bbox": [[0.193, 0.847, 0.734, 0.858]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 342, "edu_l1_label": "IOS"}, {"txt": "4.1.3Accuracy Score", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15000000, "bbox": [[0.139, 0.879, 0.323, 0.891]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 8, "global_sentence_id": 343, "edu_l1_label": "IOS"}, {"txt": "When the dataset has class labels, we assess the accuracy of the learned embedding in a classifcation task.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15100000, "bbox": [[0.139, 0.902, 0.821, 0.913]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 344, "edu_l1_label": "IOS"}, {"txt": "After performing dimensionality reduction using Consistent LLE, we train a classifer (e.g., k-Nearest Neighbors) on", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15100000, "bbox": [[0.821, 0.902, 0.861, 0.913], [0.139, 0.916, 0.861, 0.927]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 345, "edu_l1_label": "IOS"}, {"txt": "n", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15200000, "bbox": [[0.545, 0.603, 0.56, 0.612]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 28, "global_sentence_id": 346, "edu_l1_label": "EDU_O"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15200000, "bbox": [[0.491, 0.957, 0.508, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 347, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15200000, "bbox": [[0.129, 0.066, 0.87, 0.652]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 2, "global_sentence_id": 348, "edu_l1_label": "EDU_O"}, {"txt": "Figure5:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15300000, "bbox": [[0.165, 0.656, 0.473, 0.673]]}]}, "tags": ["text"], "label": "figure_title", "web_segment_id": 10, "global_sentence_id": 349, "edu_l1_label": "EDU_O"}, {"txt": "Visualizations of three datasets:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15300000, "bbox": [[0.165, 0.656, 0.473, 0.673]]}]}, "tags": ["text"], "label": "figure_title", "web_segment_id": 10, "global_sentence_id": 350, "edu_l1_label": "EDU_O"}, {"txt": "(a) Olivetti Faces, (b) MNIST, and (c) Swiss Roll.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15300000, "bbox": [[0.473, 0.656, 0.831, 0.673]]}]}, "tags": ["text"], "label": "figure_title", "web_segment_id": 9, "global_sentence_id": 352, "edu_l1_label": "EDU_O"}, {"txt": "the low-dimensional representations and compute the classifcation accuracy. The accuracy is defned as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15400000, "bbox": [[0.138, 0.699, 0.831, 0.715]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 5, "global_sentence_id": 353, "edu_l1_label": "IOS"}, {"txt": "Accuracy", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15500000, "bbox": [[0.354, 0.73, 0.432, 0.745]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 7, "global_sentence_id": 354, "edu_l1_label": "EDU_O"}, {"txt": "Number of Correct Predictions", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15500000, "bbox": [[0.435, 0.723, 0.645, 0.736]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 8, "global_sentence_id": 355, "edu_l1_label": "EDU_O"}, {"txt": "Total Number of Predictions", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15500000, "bbox": [[0.442, 0.735, 0.637, 0.752]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 6, "global_sentence_id": 356, "edu_l1_label": "EDU_O"}, {"txt": "This metric helps us evaluate whether the low-dimensional embedding preserves class separability, which is essential for downstream classifcation tasks.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15600000, "bbox": [[0.139, 0.759, 0.861, 0.77], [0.139, 0.773, 0.432, 0.784]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 357, "edu_l1_label": "IOS"}, {"txt": "4.2 Datasets and Experimental Setup", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15700000, "bbox": [[0.139, 0.805, 0.167, 0.82], [0.189, 0.805, 0.513, 0.82]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 1, "global_sentence_id": 358, "edu_l1_label": "IOS"}, {"txt": "In our experiments, we employed the Adam optimizer [9].Additionally, we evaluated the performance of the proposed ALLE algorithm using a range of well-established datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15800000, "bbox": [[0.139, 0.83, 0.861, 0.841], [0.139, 0.844, 0.621, 0.855]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 359, "edu_l1_label": "IOS"}, {"txt": "These datasets, encompassing both synthetic and real-world data, are widely recognized in the dimensionality reduction literature and serve as standard benchmarks for algorithm evaluation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15800000, "bbox": [[0.621, 0.844, 0.861, 0.855], [0.139, 0.857, 0.861, 0.869], [0.139, 0.871, 0.448, 0.883]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 360, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15900000, "bbox": [[0.492, 0.957, 0.507, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 361, "edu_l1_label": "EDU_O"}, {"txt": "table", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 15900000, "bbox": [[0.178, 0.068, 0.819, 0.26]]}]}, "tags": ["table"], "label": "table", "web_segment_id": 2, "global_sentence_id": 362, "edu_l1_label": "EDU_O"}, {"txt": "Table 1: Comparison of Standard LLE, ISO LLE and ALLE on various datasets based on Trustworthy and Continuity metrics.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16000000, "bbox": [[0.134, 0.268, 0.863, 0.298]]}]}, "tags": ["table_caption"], "label": "table_title", "web_segment_id": 0, "global_sentence_id": 363, "edu_l1_label": "EDU_O"}, {"txt": "table", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16100000, "bbox": [[0.143, 0.311, 0.856, 0.445]]}]}, "tags": ["table"], "label": "table", "web_segment_id": 1, "global_sentence_id": 364, "edu_l1_label": "EDU_O"}, {"txt": "Table 2: Comparison of Standard LLE, ISO LLE and ALLE on various datasets based on SVC, KNN, and Silhouette Score metrics.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16200000, "bbox": [[0.136, 0.453, 0.86, 0.484]]}]}, "tags": ["table_caption"], "label": "table_title", "web_segment_id": 12, "global_sentence_id": 365, "edu_l1_label": "EDU_O"}, {"txt": "4.2.1 MNIST", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16300000, "bbox": [[0.139, 0.513, 0.177, 0.526], [0.196, 0.513, 0.259, 0.526]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 11, "global_sentence_id": 366, "edu_l1_label": "IOS"}, {"txt": "The MNIST dataset [5], consisting of 70,000 grayscale images of handwritten digits (28x28 pixels), serves as a benchmark for classifcation and dimensionality reduction.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16400000, "bbox": [[0.139, 0.536, 0.861, 0.547], [0.139, 0.55, 0.563, 0.561]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 367, "edu_l1_label": "IOS"}, {"txt": "For this study, we evaluated the algorithm on two variations: the full dataset with 10 digit classes (0–9), ofering high variability and challenging classseparability, and a subset with 6 digit classes (0–5), providing a simpler task with reduced inter-class variabilitywhile maintaining the need for clear separability.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16400000, "bbox": [[0.563, 0.55, 0.861, 0.561], [0.139, 0.564, 0.861, 0.575], [0.139, 0.578, 0.861, 0.589], [0.139, 0.591, 0.461, 0.603]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 368, "edu_l1_label": "IOS"}, {"txt": "4.2.2Swiss Roll", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16500000, "bbox": [[0.139, 0.623, 0.28, 0.636]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 10, "global_sentence_id": 369, "edu_l1_label": "IOS"}, {"txt": "The Swiss Roll dataset is a popular synthetic dataset used for testing manifold learning algorithms.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16600000, "bbox": [[0.139, 0.646, 0.774, 0.657]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 370, "edu_l1_label": "IOS"}, {"txt": "It consists of points uniformly sampled from a two-dimensional manifold embedded in three-dimensional space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16600000, "bbox": [[0.774, 0.646, 0.861, 0.657], [0.139, 0.66, 0.777, 0.671]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 371, "edu_l1_label": "IOS"}, {"txt": "The dataset is challenging because it requires the algorithm to unfold the non-linear manifold into a lower-dimensional space while preserving its intrinsic structure.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16600000, "bbox": [[0.777, 0.66, 0.861, 0.671], [0.139, 0.674, 0.861, 0.685], [0.139, 0.688, 0.434, 0.699]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 372, "edu_l1_label": "IOS"}, {"txt": "4.2.3 Olivetti Faces", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16700000, "bbox": [[0.139, 0.719, 0.177, 0.732], [0.196, 0.719, 0.308, 0.732]]}]}, "tags": ["title"], "label": "title3", "web_segment_id": 9, "global_sentence_id": 373, "edu_l1_label": "IOS"}, {"txt": "The Olivetti Faces Dataset consists of 400 grayscale images of size 64 × 64 pixels, representing 40 distinctclasses (individuals) with 10 images per class.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16800000, "bbox": [[0.139, 0.742, 0.861, 0.754], [0.139, 0.756, 0.444, 0.767]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 374, "edu_l1_label": "IOS"}, {"txt": "The images capture variations in facial expressions and poses under controlled conditions, making it a widely used benchmark for tasks such as dimensionality reduction, face recognition, and clustering.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16800000, "bbox": [[0.444, 0.756, 0.861, 0.767], [0.139, 0.77, 0.861, 0.781], [0.139, 0.784, 0.349, 0.795]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 375, "edu_l1_label": "IOS"}, {"txt": "4.3 Compare Results", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 16900000, "bbox": [[0.139, 0.816, 0.167, 0.831], [0.189, 0.816, 0.354, 0.831]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 8, "global_sentence_id": 376, "edu_l1_label": "IOS"}, {"txt": "In this section, we provide a comprehensive comparative analysis of the proposed ALLE algorithm against the standard LLE approach.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17000000, "bbox": [[0.139, 0.841, 0.861, 0.852], [0.139, 0.855, 0.3, 0.866]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 377, "edu_l1_label": "IOS"}, {"txt": "The evaluation metrics utilized in this comparison are introduced in Section 4.1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17000000, "bbox": [[0.3, 0.855, 0.837, 0.866]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 378, "edu_l1_label": "IOS"}, {"txt": "The results clearly demonstrate the efectiveness of the adaptive metric in preserving neighborhood structures and improving the quality of low-dimensional embeddings.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.163, 0.869, 0.861, 0.88], [0.139, 0.883, 0.551, 0.894]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 379, "edu_l1_label": "IOS"}, {"txt": "As illustrated in Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.551, 0.883, 0.69, 0.894]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 380, "edu_l1_label": "IOS"}, {"txt": "6, the diferences between Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.69, 0.883, 0.861, 0.894], [0.139, 0.896, 0.165, 0.908]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 381, "edu_l1_label": "IOS"}, {"txt": "6a and Fig.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.165, 0.896, 0.248, 0.908]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 382, "edu_l1_label": "IOS"}, {"txt": "6b highlight the capability of the ALLE algorithm to adaptively stretch and compress data in multiple directions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.248, 0.896, 0.861, 0.908], [0.139, 0.91, 0.284, 0.922]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 383, "edu_l1_label": "IOS"}, {"txt": "This fexibility leads to signifcant improvements in trustworthiness and continuity, as", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17100000, "bbox": [[0.284, 0.91, 0.861, 0.922]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 384, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17200000, "bbox": [[0.489, 0.957, 0.509, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 385, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17200000, "bbox": [[0.139, 0.07, 0.463, 0.265]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 1, "global_sentence_id": 386, "edu_l1_label": "EDU_O"}, {"txt": "(a) Olivetti using Standard LLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17300000, "bbox": [[0.194, 0.269, 0.413, 0.285]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 10, "global_sentence_id": 387, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17400000, "bbox": [[0.14, 0.287, 0.464, 0.511]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 4, "global_sentence_id": 388, "edu_l1_label": "EDU_O"}, {"txt": "(c) MNIST 10 Classes using Standard LLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17400000, "bbox": [[0.158, 0.517, 0.447, 0.533]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 8, "global_sentence_id": 389, "edu_l1_label": "EDU_O"}, {"txt": "0", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17500000, "bbox": [[0.157, 0.59, 0.165, 0.596]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 14, "global_sentence_id": 390, "edu_l1_label": "EDU_O"}, {"txt": "(e) MNIST 6 Classes using Standard LLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17500000, "bbox": [[0.16, 0.781, 0.444, 0.796]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 13, "global_sentence_id": 391, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17600000, "bbox": [[0.537, 0.072, 0.857, 0.265]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 2, "global_sentence_id": 392, "edu_l1_label": "EDU_O"}, {"txt": "(b) Olivetti Classes using", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17700000, "bbox": [[0.59, 0.269, 0.807, 0.284]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 6, "global_sentence_id": 393, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17800000, "bbox": [[0.537, 0.287, 0.848, 0.512]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 0, "global_sentence_id": 394, "edu_l1_label": "EDU_O"}, {"txt": "(d) MNIST 10 Classes ALLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17800000, "bbox": [[0.601, 0.518, 0.798, 0.532]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 12, "global_sentence_id": 395, "edu_l1_label": "EDU_O"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17900000, "bbox": [[0.532, 0.55, 0.861, 0.78]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 5, "global_sentence_id": 396, "edu_l1_label": "EDU_O"}, {"txt": "(f) MNIST 6 Classes using ALLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 17900000, "bbox": [[0.586, 0.781, 0.814, 0.795]]}]}, "tags": ["figure_caption"], "label": "O", "web_segment_id": 9, "global_sentence_id": 397, "edu_l1_label": "EDU_O"}, {"txt": "Figure 6: Results on Three Datasets Using Standard LLE and ALLE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18000000, "bbox": [[0.249, 0.807, 0.75, 0.824]]}]}, "tags": ["figure_caption"], "label": "figure_title", "web_segment_id": 3, "global_sentence_id": 398, "edu_l1_label": "EDU_O"}, {"txt": "evidenced in Table 1. Furthermore, the proposed algorithm enhances classifcation accuracy when evaluated using the Support Vector Classifer (SVC) algorithm, as detailed in Table 2. Notably, while the KNN score for standard LLE shows slightly better results, this can be attributed to the simpler geometric structure of its output, which relies purely on Euclidean distance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18100000, "bbox": [[0.137, 0.85, 0.867, 0.904]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 7, "global_sentence_id": 399, "edu_l1_label": "IOS"}, {"txt": "Similarly, Fig.6c and Fig.6d underscore the advantages of leveraging a learned adaptive metric for", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18200000, "bbox": [[0.158, 0.907, 0.863, 0.922]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 11, "global_sentence_id": 400, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.491, 0.956, 0.509, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 401, "edu_l1_label": "EDU_O"}, {"txt": "bending the topological space.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.139, 0.076, 0.34, 0.087]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 402, "edu_l1_label": "IOS"}, {"txt": "This approach yields superior visualizations in lower dimensions, particularly for complex datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.34, 0.076, 0.861, 0.087], [0.139, 0.09, 0.278, 0.101]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 403, "edu_l1_label": "IOS"}, {"txt": "For instance, the inter-class and intra-class relationships between digits such as “2” and“6” and “3” and “9” in the MNIST dataset are more distinctly represented.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.278, 0.09, 0.861, 0.101], [0.139, 0.103, 0.632, 0.115]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 404, "edu_l1_label": "IOS"}, {"txt": "Table 2 further corroborates theseobservations, demonstrating improved silhouette scores under the ALLE algorithm.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.632, 0.103, 0.861, 0.115], [0.139, 0.117, 0.682, 0.129]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 405, "edu_l1_label": "IOS"}, {"txt": "Additionally, classifcation accuracy using the SVC algorithm is notably enhanced for both the standard MNIST dataset and the reduced MNIST dataset with 10 classes.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18300000, "bbox": [[0.682, 0.117, 0.861, 0.129], [0.139, 0.131, 0.861, 0.142], [0.139, 0.145, 0.348, 0.156]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 406, "edu_l1_label": "IOS"}, {"txt": "As shown in Table 1, the continuity and trustworthiness of the Swiss roll and scaled Swiss roll datasets have shown slight improvement.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18400000, "bbox": [[0.163, 0.159, 0.861, 0.17], [0.139, 0.173, 0.315, 0.184]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 407, "edu_l1_label": "IOS"}, {"txt": "The results indicate that the visualization of ALLE is indistinguishable from that of standard LLE on this dataset.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18400000, "bbox": [[0.315, 0.173, 0.861, 0.184], [0.139, 0.186, 0.356, 0.198]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 408, "edu_l1_label": "IOS"}, {"txt": "This outcome can be attributed to the sufciency of the Euclidean distance in the linear reconstruction process for this dataset, rendering the metric modifcations inconsequential.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18400000, "bbox": [[0.356, 0.186, 0.861, 0.198], [0.139, 0.2, 0.822, 0.212]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 409, "edu_l1_label": "IOS"}, {"txt": "Interestingly, while standard LLE achieves better continuity scores for specifc datasets, this outcome can be attributed to the bending of the topological space by ALLE.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18500000, "bbox": [[0.163, 0.214, 0.861, 0.225], [0.139, 0.228, 0.548, 0.239]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 410, "edu_l1_label": "IOS"}, {"txt": "This adaptation creates a clearer separation between data points from diferent classes, which, under the initial KNN-based neighbor selection, may otherwise have been incorrectly classifed as neighbors.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18500000, "bbox": [[0.548, 0.228, 0.861, 0.239], [0.139, 0.242, 0.861, 0.253], [0.139, 0.256, 0.433, 0.267]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 411, "edu_l1_label": "IOS"}, {"txt": "These fndings confrm that the ALLE algorithm efectively balances the trade-of between improving global visualization quality and maintaining local neighborhood structures, making it a powerful tool for dimensionality reduction and data analysis.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18600000, "bbox": [[0.163, 0.269, 0.861, 0.281], [0.139, 0.283, 0.861, 0.295], [0.139, 0.297, 0.426, 0.308]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 412, "edu_l1_label": "IOS"}, {"txt": "4.4Conclusion and Future Work", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18700000, "bbox": [[0.139, 0.329, 0.468, 0.345]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 6, "global_sentence_id": 413, "edu_l1_label": "IOS"}, {"txt": "To wrap up, the ALLE algorithm has shown clear advantages over the standard LLE approach.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18800000, "bbox": [[0.139, 0.354, 0.776, 0.366]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 414, "edu_l1_label": "IOS"}, {"txt": "It preserves neighborhood structures more efectively, creates higher-quality low-dimensional embeddings, and improves classifcation accuracy.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18800000, "bbox": [[0.776, 0.354, 0.861, 0.366], [0.139, 0.368, 0.861, 0.379], [0.139, 0.382, 0.286, 0.393]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 415, "edu_l1_label": "IOS"}, {"txt": "By adapting the metric to stretch and compress the data, ALLE opens new possibilities for analyzing and visualizing complex datasets.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18800000, "bbox": [[0.286, 0.382, 0.861, 0.393], [0.139, 0.396, 0.456, 0.407]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 416, "edu_l1_label": "IOS"}, {"txt": "However, some trade-ofs, such as slightly lower continuity scores in certain cases, provide areas for further refnement.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18800000, "bbox": [[0.456, 0.396, 0.861, 0.407], [0.139, 0.41, 0.532, 0.421]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 417, "edu_l1_label": "IOS"}, {"txt": "Looking ahead, there are exciting opportunities to build on this work.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18900000, "bbox": [[0.163, 0.423, 0.615, 0.435]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 418, "edu_l1_label": "IOS"}, {"txt": "Future eforts could focus on improving the optimization process for the adaptive metric to strike an even better balance between trustworthiness and continuity.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18900000, "bbox": [[0.615, 0.423, 0.861, 0.435], [0.139, 0.437, 0.861, 0.449], [0.139, 0.451, 0.236, 0.462]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 419, "edu_l1_label": "IOS"}, {"txt": "Scaling the algorithm for larger datasets and integrating it with deep learning frameworks could make it even more powerful and versatile.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18900000, "bbox": [[0.236, 0.451, 0.861, 0.462], [0.139, 0.465, 0.414, 0.476]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 420, "edu_l1_label": "IOS"}, {"txt": "Exploring real-time applications where the adaptive metric evolves dynamically with the data is another promising avenue.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18900000, "bbox": [[0.414, 0.465, 0.861, 0.476], [0.139, 0.479, 0.508, 0.49]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 421, "edu_l1_label": "IOS"}, {"txt": "These steps will help solidify ALLE as a method for dimensionality reduction and visualization in increasingly complex data scenarios.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 18900000, "bbox": [[0.508, 0.479, 0.861, 0.49], [0.139, 0.493, 0.68, 0.504]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 422, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19000000, "bbox": [[0.492, 0.956, 0.509, 0.968]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 423, "edu_l1_label": "EDU_O"}, {"txt": "AProofs and Derivations", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19100000, "bbox": [[0.139, 0.074, 0.451, 0.092]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 5, "global_sentence_id": 424, "edu_l1_label": "BOS"}, {"txt": "A.1 Metric Optimization", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19200000, "bbox": [[0.139, 0.109, 0.173, 0.124], [0.195, 0.109, 0.392, 0.124]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 3, "global_sentence_id": 425, "edu_l1_label": "IOS"}, {"txt": "Here, we provide the detailed derivation of the objective function for metric learning.Starting from the objective function:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19300000, "bbox": [[0.139, 0.134, 0.861, 0.145], [0.139, 0.148, 0.261, 0.159]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 426, "edu_l1_label": "IOS"}, {"txt": "E ( \\mathbf { M } ) = \\sum _ { i = 1 } ^ { N } \\left\\| \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { x } _ { j } \\right\\| _ { \\mathbf { M } } ^ { 2 } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19400000, "bbox": [[0.376, 0.156, 0.618, 0.202]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 25, "global_sentence_id": 427, "edu_l1_label": "IOS"}, {"txt": "expanding this, we have:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19500000, "bbox": [[0.139, 0.206, 0.301, 0.218]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 428, "edu_l1_label": "IOS"}, {"txt": "E ( \\mathbf { M } ) = \\sum _ { i = 1 } ^ { N } \\left( \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { x } _ { j } \\right) ^ { T } \\mathbf { M } ( \\mathbf { x } _ { i } - \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { x } _ { j } ) .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19600000, "bbox": [[0.313, 0.222, 0.685, 0.262]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 23, "global_sentence_id": 429, "edu_l1_label": "IOS"}, {"txt": "Defning ri = xi −Pj∈N (i) w˜ijxj, we substitute to get:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19700000, "bbox": [[0.139, 0.271, 0.505, 0.282]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 430, "edu_l1_label": "IOS"}, {"txt": "E ( \\mathbf { M } ) = \\sum _ { i = 1 } ^ { N } \\mathbf { r } _ { i } ^ { T } \\mathbf { M } \\mathbf { r } _ { i } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19800000, "bbox": [[0.426, 0.288, 0.565, 0.327]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 26, "global_sentence_id": 431, "edu_l1_label": "IOS"}, {"txt": "Then, diferentiating E(M) with respect to M gives:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 19900000, "bbox": [[0.163, 0.336, 0.511, 0.347]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 12, "global_sentence_id": 432, "edu_l1_label": "IOS"}, {"txt": "\\frac { \\partial E ( \\mathbf { M } ) } { \\partial \\mathbf { M } } = \\sum _ { i = 1 } ^ { N } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T } .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20000000, "bbox": [[0.431, 0.352, 0.565, 0.388]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 24, "global_sentence_id": 433, "edu_l1_label": "IOS"}, {"txt": "This result allows us to update M(t+1) using gradient descent:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20100000, "bbox": [[0.139, 0.4, 0.551, 0.412]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 434, "edu_l1_label": "IOS"}, {"txt": "\\mathbf { M } ^ { ( t + 1 ) } = \\mathbf { M } ^ { ( t ) } - \\eta \\sum _ { i = 1 } ^ { N } \\mathbf { r } _ { i } \\mathbf { r } _ { i } ^ { T } .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20200000, "bbox": [[0.4, 0.415, 0.591, 0.454]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 28, "global_sentence_id": 435, "edu_l1_label": "IOS"}, {"txt": "A.2 Linear Embedding", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20300000, "bbox": [[0.139, 0.473, 0.173, 0.488], [0.195, 0.473, 0.372, 0.488]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 2, "global_sentence_id": 436, "edu_l1_label": "IOS"}, {"txt": "The linear embedding optimization problem is formulated as:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20400000, "bbox": [[0.139, 0.498, 0.543, 0.509]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 437, "edu_l1_label": "IOS"}, {"txt": "\\operatorname* { m i n } _ { \\mathbf { Y } } \\sum _ { i = 1 } ^ { N } \\left\\| \\mathbf { y } _ { i } = \\sum _ { j \\in \\mathcal { N } ( i ) } \\mathbf { \\tilde { w } } _ { i j } \\mathbf { y } _ { j } \\right\\| _ { 2 } ^ { 2 } ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20500000, "bbox": [[0.396, 0.514, 0.596, 0.561]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 32, "global_sentence_id": 438, "edu_l1_label": "IOS"}, {"txt": "with the constraints:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20600000, "bbox": [[0.139, 0.567, 0.275, 0.579]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 13, "global_sentence_id": 439, "edu_l1_label": "IOS"}, {"txt": "\\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } { { \\bf { y } } _ { i } { \\bf { y } } _ { i } ^ { T } } = { \\bf { I } } , \\quad \\sum _ { i = 1 } ^ { n } { { { \\bf { y } } _ { i } } } = { \\bf { 0 } } .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20700000, "bbox": [[0.396, 0.572, 0.598, 0.61]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 29, "global_sentence_id": 440, "edu_l1_label": "IOS"}, {"txt": "Expanding the objective function, we have:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20800000, "bbox": [[0.163, 0.614, 0.448, 0.625]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 11, "global_sentence_id": 441, "edu_l1_label": "IOS"}, {"txt": "\\operatorname* { m i n } _ { \\mathbf { Y } } \\sum _ { i = 1 } ^ { N } \\left( \\mathbf { y } _ { i } - \\sum _ { j = 1 } ^ { n } \\mathbf { w } _ { i j } \\mathbf { y } _ { j } \\right) ^ { T } \\left( \\mathbf { y } _ { i } - \\sum _ { j = 1 } ^ { n } \\mathbf { w } _ { i j } \\mathbf { y } _ { j } \\right) .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 20900000, "bbox": [[0.337, 0.628, 0.654, 0.672]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 27, "global_sentence_id": 442, "edu_l1_label": "IOS"}, {"txt": "In matrix form, this becomes:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21000000, "bbox": [[0.139, 0.678, 0.335, 0.689]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 14, "global_sentence_id": 443, "edu_l1_label": "IOS"}, {"txt": "min tr (YT(I - W)”(I - W)Y)", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21100000, "bbox": [[0.385, 0.688, 0.609, 0.71]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 22, "global_sentence_id": 444, "edu_l1_label": "IOS"}, {"txt": "Defning MW = (I −W)T (I −W), we can simplify to:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21200000, "bbox": [[0.139, 0.717, 0.502, 0.728]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 16, "global_sentence_id": 445, "edu_l1_label": "IOS"}, {"txt": "To solve this using Lagrange multipliers, defne the Lagrangian:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21300000, "bbox": [[0.163, 0.765, 0.583, 0.777]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 7, "global_sentence_id": 446, "edu_l1_label": "IOS"}, {"txt": "L ( { \\bf Y } , \\Lambda ) = \\mathrm { t r } ( { \\bf Y } ^ { T } { \\bf M } _ { W } { \\bf Y } ) - \\mathrm { t r } \\left( \\Lambda \\left( \\frac { 1 } { n } { \\bf Y } ^ { T } { \\bf Y } - { \\bf I } \\right) \\right) .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21400000, "bbox": [[0.325, 0.781, 0.667, 0.81]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 30, "global_sentence_id": 447, "edu_l1_label": "IOS"}, {"txt": "Taking the derivative with respect to Y:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21500000, "bbox": [[0.139, 0.819, 0.406, 0.83]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 10, "global_sentence_id": 448, "edu_l1_label": "IOS"}, {"txt": "\\frac { \\partial L } { \\partial { \\bf Y } } = 2 { \\bf M } _ { W } { \\bf Y } - 2 \\frac { 1 } { n } { \\bf Y } \\Lambda = 0 .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21600000, "bbox": [[0.396, 0.833, 0.596, 0.862]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 31, "global_sentence_id": 449, "edu_l1_label": "IOS"}, {"txt": "This leads to an eigenvalue problem:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21700000, "bbox": [[0.139, 0.868, 0.38, 0.879]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 15, "global_sentence_id": 450, "edu_l1_label": "IOS"}, {"txt": "MWY=", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21800000, "bbox": [[0.448, 0.885, 0.5, 0.898]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 19, "global_sentence_id": 451, "edu_l1_label": "IOS"}, {"txt": "or equivalently:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 21900000, "bbox": [[0.139, 0.906, 0.24, 0.917]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 8, "global_sentence_id": 452, "edu_l1_label": "EDU_O"}, {"txt": "一", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22000000, "bbox": [[0.514, 0.88, 0.524, 0.89]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 21, "global_sentence_id": 453, "edu_l1_label": "EDU_O"}, {"txt": "一", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22000000, "bbox": [[0.496, 0.888, 0.504, 0.894]]}]}, "tags": ["text"], "label": "O", "web_segment_id": 18, "global_sentence_id": 454, "edu_l1_label": "EDU_O"}, {"txt": "YA,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22000000, "bbox": [[0.521, 0.882, 0.552, 0.899]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 20, "global_sentence_id": 455, "edu_l1_label": "IOS"}, {"txt": "MWY = XY.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22000000, "bbox": [[0.453, 0.918, 0.545, 0.933]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 17, "global_sentence_id": 456, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22100000, "bbox": [[0.491, 0.955, 0.51, 0.969]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 457, "edu_l1_label": "EDU_O"}, {"txt": "References", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22200000, "bbox": [[0.139, 0.074, 0.262, 0.092]]}]}, "tags": ["title"], "label": "reference", "web_segment_id": 1, "global_sentence_id": 458, "edu_l1_label": "EDU_O"}, {"txt": "[1] Mikhail Belkin and Partha Niyogi.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.108, 0.399, 0.119]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 459, "edu_l1_label": "EDU_O"}, {"txt": "Laplacian eigenmaps for dimensionality reduction and data representation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.399, 0.108, 0.861, 0.119], [0.171, 0.122, 0.214, 0.133]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 460, "edu_l1_label": "EDU_O"}, {"txt": "Neural computation, 15(6):1373–1396, 2003.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.214, 0.122, 0.509, 0.133]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 461, "edu_l1_label": "EDU_O"}, {"txt": "[2] Christopher M. Bishop.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.139, 0.328, 0.15]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 462, "edu_l1_label": "EDU_O"}, {"txt": "Pattern Recognition and Machine Learning.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.328, 0.139, 0.632, 0.15]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 463, "edu_l1_label": "EDU_O"}, {"txt": "Springer, New York, 1st edition, 2006.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.632, 0.139, 0.861, 0.15], [0.171, 0.153, 0.205, 0.164]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 464, "edu_l1_label": "EDU_O"}, {"txt": "[3] Hong Chang and Dit-Yan Yeung.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.17, 0.387, 0.182]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 465, "edu_l1_label": "EDU_O"}, {"txt": "Robust locally linear embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.387, 0.17, 0.607, 0.182]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 466, "edu_l1_label": "EDU_O"}, {"txt": "Pattern Recognition, 39(6):1053–1065,2006.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.607, 0.17, 0.861, 0.182], [0.171, 0.184, 0.205, 0.195]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 467, "edu_l1_label": "EDU_O"}, {"txt": "[4] Jing Chen and Yang Liu.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.201, 0.336, 0.213]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 468, "edu_l1_label": "EDU_O"}, {"txt": "Locally linear embedding: a survey.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.336, 0.201, 0.576, 0.213]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 469, "edu_l1_label": "EDU_O"}, {"txt": "Artifcial Intelligence Review, 36(1):29–48,2011.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.576, 0.201, 0.861, 0.213], [0.171, 0.215, 0.205, 0.227]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 470, "edu_l1_label": "EDU_O"}, {"txt": "[5] Li Deng.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.233, 0.228, 0.244]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 471, "edu_l1_label": "EDU_O"}, {"txt": "The mnist database of handwritten digit images for machine learning research.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.228, 0.233, 0.769, 0.244]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 472, "edu_l1_label": "EDU_O"}, {"txt": "IEEE Signal Processing Magazine, 29(6):141–142, 2012.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.769, 0.233, 0.861, 0.244], [0.171, 0.246, 0.45, 0.258]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 473, "edu_l1_label": "EDU_O"}, {"txt": "[6] R. A. Fisher.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.264, 0.256, 0.275]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 474, "edu_l1_label": "EDU_O"}, {"txt": "Iris.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.256, 0.264, 0.289, 0.275]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 475, "edu_l1_label": "EDU_O"}, {"txt": "UCI Machine Learning Repository, 1936.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.289, 0.264, 0.565, 0.275]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 476, "edu_l1_label": "EDU_O"}, {"txt": "DOI: https://doi.org/10.24432/C56C76.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.565, 0.264, 0.836, 0.275]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 477, "edu_l1_label": "EDU_O"}, {"txt": "[7] Ali Ghodsi.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.281, 0.247, 0.292]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 478, "edu_l1_label": "EDU_O"}, {"txt": "Dimensionality reduction: a short tutorial.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.247, 0.281, 0.541, 0.292]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 479, "edu_l1_label": "EDU_O"}, {"txt": "Technical report, Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada, 2006.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.541, 0.281, 0.861, 0.292], [0.171, 0.295, 0.675, 0.306]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 480, "edu_l1_label": "EDU_O"}, {"txt": "[8] Harold Hotelling.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.312, 0.286, 0.324]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 481, "edu_l1_label": "EDU_O"}, {"txt": "Analysis of a complex of statistical variables into principal components.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.286, 0.312, 0.782, 0.324]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 482, "edu_l1_label": "EDU_O"}, {"txt": "Journal of Educational Psychology, 1933.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.782, 0.312, 0.861, 0.324], [0.171, 0.326, 0.367, 0.337]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 483, "edu_l1_label": "EDU_O"}, {"txt": "[9] Diederik P Kingma and Jimmy Ba.Adam:A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.146, 0.343, 0.861, 0.355], [0.171, 0.357, 0.321, 0.369]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 484, "edu_l1_label": "EDU_O"}, {"txt": "[10] Yuanhong Liu, Baoxin Shi, Shixiang Lu, Zhi-Wei Gao, and Fangfang Zhang.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.375, 0.66, 0.386]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 485, "edu_l1_label": "EDU_O"}, {"txt": "A novel local linear embedding algorithm via local mutual representation for bearing fault diagnosis.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.66, 0.375, 0.861, 0.386], [0.171, 0.388, 0.625, 0.4]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 486, "edu_l1_label": "EDU_O"}, {"txt": "Reliability Engineering and System Safety, 247:110135, 2024.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.625, 0.388, 0.861, 0.4], [0.171, 0.402, 0.335, 0.414]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 487, "edu_l1_label": "EDU_O"}, {"txt": "[11] James R. Munkres.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.42, 0.297, 0.431]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 488, "edu_l1_label": "EDU_O"}, {"txt": "Topology.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.297, 0.42, 0.365, 0.431]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 489, "edu_l1_label": "EDU_O"}, {"txt": "Prentice Hall, Upper Saddle River, NJ, 2nd edition, 2000.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.365, 0.42, 0.753, 0.431]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 490, "edu_l1_label": "EDU_O"}, {"txt": "[12] Yaozhang Pan, Shuzhi Ge, and Abdullah Al-Mamun.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.437, 0.526, 0.448]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 491, "edu_l1_label": "EDU_O"}, {"txt": "Weighted locally linear embedding for dimension reduction.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.526, 0.437, 0.861, 0.448], [0.171, 0.451, 0.237, 0.462]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 492, "edu_l1_label": "EDU_O"}, {"txt": "Pattern Recognition, 42:798–811, 05 2009.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.237, 0.451, 0.518, 0.462]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 493, "edu_l1_label": "EDU_O"}, {"txt": "[13] Peter J. Rousseeuw.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.468, 0.303, 0.479]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 494, "edu_l1_label": "EDU_O"}, {"txt": "Silhouettes: A graphical aid to the interpretation and validation of cluster analysis.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.303, 0.468, 0.861, 0.479]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 495, "edu_l1_label": "EDU_O"}, {"txt": "Journal of Computational and Applied Mathematics, 20:53–65, 1987.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.171, 0.482, 0.622, 0.493]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 496, "edu_l1_label": "EDU_O"}, {"txt": "[14] Sam T Roweis and Lawrence K Saul.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.499, 0.422, 0.511]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 497, "edu_l1_label": "EDU_O"}, {"txt": "Nonlinear dimensionality reduction by locally linear embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.422, 0.499, 0.861, 0.511]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 498, "edu_l1_label": "EDU_O"}, {"txt": "Science, 290(5500):2323–2326, 2000.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.171, 0.513, 0.408, 0.524]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 499, "edu_l1_label": "EDU_O"}, {"txt": "[15] Yunting Tao, Yuqun Li, Fanyu Kong, Yuliang Shi, Ming Yang, Jia Yu, and Hanlin Zhang.Privacypreserving outsourcing scheme of face recognition based on locally linear embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.53, 0.861, 0.542], [0.171, 0.544, 0.727, 0.556]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 500, "edu_l1_label": "EDU_O"}, {"txt": "Computers & Security, 144:103931, 2024.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.727, 0.544, 0.861, 0.556], [0.171, 0.558, 0.318, 0.569]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 501, "edu_l1_label": "EDU_O"}, {"txt": "[16] Joshua B Tenenbaum, Vin de Silva, and John C Langford.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.575, 0.562, 0.587]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 502, "edu_l1_label": "EDU_O"}, {"txt": "A global geometric framework for nonlinear dimensionality reduction.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.562, 0.575, 0.861, 0.587], [0.171, 0.589, 0.337, 0.601]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 503, "edu_l1_label": "EDU_O"}, {"txt": "Science, 290(5500):2319–2323, 2000.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.337, 0.589, 0.581, 0.601]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 504, "edu_l1_label": "EDU_O"}, {"txt": "[17] Warren S Torgerson.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.607, 0.307, 0.618]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 505, "edu_l1_label": "EDU_O"}, {"txt": "Multidimensional scaling: I. theory and method.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.307, 0.607, 0.633, 0.618]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 506, "edu_l1_label": "EDU_O"}, {"txt": "Psychometrika, 1952.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.633, 0.607, 0.779, 0.618]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 507, "edu_l1_label": "EDU_O"}, {"txt": "[18] Claudio Varini, Andreas Degenhard, and Tim Wilhelm Nattkemper.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.624, 0.612, 0.635]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 508, "edu_l1_label": "EDU_O"}, {"txt": "ISOLLE: LLE with geodesic distance.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.612, 0.624, 0.861, 0.635]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 509, "edu_l1_label": "EDU_O"}, {"txt": "Neurocomputing, 69:1768–1771, 2006.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.171, 0.638, 0.416, 0.649]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 510, "edu_l1_label": "EDU_O"}, {"txt": "[19] Jarkko Venna and Samuel Kaski.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.655, 0.387, 0.666]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 511, "edu_l1_label": "EDU_O"}, {"txt": "Neighborhood preservation in nonlinear projection methods: An experimental study.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.387, 0.655, 0.861, 0.666], [0.171, 0.669, 0.265, 0.68]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 512, "edu_l1_label": "EDU_O"}, {"txt": "In Proceedings of the International Conference on Artifcial Neural Networks (ICANN), pages 485–491.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.265, 0.669, 0.861, 0.68], [0.171, 0.683, 0.268, 0.694]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 513, "edu_l1_label": "EDU_O"}, {"txt": "Springer, 2001.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.268, 0.683, 0.372, 0.694]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 514, "edu_l1_label": "EDU_O"}, {"txt": "[20] Chenchen Wang, Zhichen Gu, and Jin-Mao Wei.Spectral clustering and embedding with inter-class topology-preserving.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.7, 0.861, 0.711], [0.171, 0.714, 0.304, 0.725]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 515, "edu_l1_label": "EDU_O"}, {"txt": "Knowledge-Based Systems, 284:111278, 2024.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.304, 0.714, 0.607, 0.725]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 516, "edu_l1_label": "EDU_O"}, {"txt": "[21] Kilian Q Weinberger and Lawrence K Saul.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.731, 0.461, 0.743]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 517, "edu_l1_label": "EDU_O"}, {"txt": "Distance metric learning for large margin nearest neighbor classifcation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.461, 0.731, 0.861, 0.743], [0.171, 0.745, 0.258, 0.756]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 518, "edu_l1_label": "EDU_O"}, {"txt": "Journal of Machine Learning Research, 10(Feb):207–244, 2009.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.258, 0.745, 0.679, 0.756]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 519, "edu_l1_label": "EDU_O"}, {"txt": "[22] Guihua Wen, Lijun Jiang, and Jun Wen.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.762, 0.442, 0.774]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 520, "edu_l1_label": "EDU_O"}, {"txt": "Kernel relative transformation with applications to enhancing locally linear embedding.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.442, 0.762, 0.861, 0.774], [0.171, 0.776, 0.334, 0.788]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 521, "edu_l1_label": "EDU_O"}, {"txt": "In 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence), pages 3401–3406, 2008.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.334, 0.776, 0.861, 0.788], [0.171, 0.79, 0.6, 0.801]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 522, "edu_l1_label": "EDU_O"}, {"txt": "[23] Stephen Willard.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.807, 0.282, 0.819]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 523, "edu_l1_label": "EDU_O"}, {"txt": "General Topology.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.282, 0.807, 0.406, 0.819]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 524, "edu_l1_label": "EDU_O"}, {"txt": "Dover Publications, 2004.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.406, 0.807, 0.582, 0.819]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 525, "edu_l1_label": "EDU_O"}, {"txt": "[24] Eric P Xing, Andrew Y Ng, Michael I Jordan, and Stuart J Russell.Distance metric learning with application to clustering with side-information.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.825, 0.861, 0.836], [0.171, 0.839, 0.486, 0.85]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 526, "edu_l1_label": "EDU_O"}, {"txt": "In Advances in neural information processing systems, pages 505–512, 2003.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.486, 0.839, 0.861, 0.85], [0.171, 0.852, 0.308, 0.864]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 527, "edu_l1_label": "EDU_O"}, {"txt": "[25] Yichen Xu and Eric Li.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.87, 0.326, 0.881]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 528, "edu_l1_label": "EDU_O"}, {"txt": "Robust locally nonlinear embedding (rlne) for dimensionality reduction of highdimensional data with noise.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.326, 0.87, 0.861, 0.881], [0.171, 0.884, 0.359, 0.895]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 529, "edu_l1_label": "EDU_O"}, {"txt": "Neurocomputing, 596:127900, 2024.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.359, 0.884, 0.596, 0.895]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 530, "edu_l1_label": "EDU_O"}, {"txt": "[26] Zhenyue Zhang and Hongyuan Zha.Principal manifolds and nonlinear dimensionality reduction via tangent space alignment.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.139, 0.901, 0.861, 0.912], [0.171, 0.915, 0.335, 0.926]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 531, "edu_l1_label": "EDU_O"}, {"txt": "SIAM Journal on Scientifc Computing, 26(1):313–338, 2007.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22300000, "bbox": [[0.335, 0.915, 0.744, 0.926]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 0, "global_sentence_id": 532, "edu_l1_label": "EDU_O"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 22400000, "bbox": [[0.491, 0.957, 0.509, 0.967]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 533, "edu_l1_label": "EDU_O"}], "type": "PDF"}
# Adaptive Locally Linear Embedding
## 1Introduction
### 1.1 Motivation
### 1.2Topological Space and Neighborhoods
### 1.3The Metric Topology
### 1.4Innovation: Need for a Flexible Metric
### 1.5Manifold Perspective
## 2Related Work
## 3The proposed method
### 3.1Notations and defnitions
### 3.2Problem Formulation
### 3.3Initialization
#### 3.3.1Find Nearest Neighbors
#### 3.3.2Initialize Metric
### 3.4Optimization Problem
#### 3.4.1Reconstruction
#### 3.4.2 Metric Optimization
#### 3.4.3 Linear Embedding
### 3.5Dimension-Reduced Data
### 3.6Theoretical Guarantee: Ensuring Positive Semi-Defniteness of M
## 4Experimental results
### 4.1Evaluation Metrics
#### 4.1.1Neighborhood Preservation
##### • Trustworthiness quantifes how many of the original nearest neighbors in high-dimensional space areretained as neighbors in the low-dimensional space.
##### • Continuity evaluates how well the low-dimensional neighbors are true neighbors in the original space.Continuity is computed as:
#### 4.1.2Silhouette Score
#### 4.1.3Accuracy Score
### 4.2 Datasets and Experimental Setup
#### 4.2.1 MNIST
#### 4.2.2Swiss Roll
#### 4.2.3 Olivetti Faces
### 4.3 Compare Results
### 4.4Conclusion and Future Work
## AProofs and Derivations
### A.1 Metric Optimization
### A.2 Linear Embedding
