# Self-supervised Speaker Diarization

###### Abstract

Over the last few years, deep learning has grown in popularity for speaker
verification, identification, and diarization. Inarguably, a significant part
of this success is due to the demonstrated effectiveness of their speaker
representations. These, however, are heavily dependent on large amounts of
annotated data and can be sensitive to new domains. This study proposes an
entirely unsupervised deep-learning model for speaker diarization.
Specifically, the study focuses on generating high-quality neural speaker
representations without any annotated data, as well as on estimating secondary
hyperparameters of the model without annotations.

The speaker embeddings are represented by an encoder trained in a self-
supervised fashion using pairs of adjacent segments assumed to be of the same
speaker. The trained encoder model is then used to self-generate pseudo-labels
to subsequently train a similarity score between different segments of the
same call using probabilistic linear discriminant analysis (PLDA) and further
to learn a clustering stopping threshold. We compared our model to state-of-
the-art unsupervised as well as supervised baselines on the CallHome
benchmarks. According to empirical results, our approach outperforms
unsupervised methods when only two speakers are present in the call, and is
only slightly worse than recent supervised models.

Index Terms: speaker diarization, self-supervised training, unsupervised PLDA

##  1 Introduction

Speaker diarization is the task of determining “who spoke when” in an audio
recording, usually with an unknown number of speakers and variable speech
duration. Traditionally, speaker diarization systems consist of four
submodules. The first module extracts speech segments using a voice activity
detector (VAD). The second module transforms the spoken segments into fixed-
length vectors representing speakers’ identity. This transformation is
performed using a parametric probabilistic model, such as i-vectors based on
GMM-UBM [1], or using deep neural network (DNN) embeddings, such as x-vectors
[2] or d-vectors [3]. The third module finds a pairwise similarity measurement
between segments using cosine distance, probabilistic linear discriminant
analysis (PLDA) [4], or DNN models [5]. Finally, the resulting similarity is
clustered to determine the number of speakers and assign a speaker label to
each segment. This is often done using a clustering model such as
agglomerative hierarchical clustering (AHC) [6], with a stopping threshold
tuned on some held-out data [7].

One common theme amongst all the methods is the need for large amounts of
labeled data, especially when considering DNN-based speaker embeddings. This
would not be an issue if not for the domain sensitivity of the diarization
systems. That is, training embeddings on one dataset might not translate into
good representations on a different domain, and annotating data in the new
domain can be prohibitively expensive.

Our study aims to build the traditional diarization pipeline, but to replace
each component with components that are trained completely unsupervised, thus
removing the need for any annotated data. This opens new opportunities in
applying speaker diarization especially for domains for which annotated data
is not available. Hence, the work is focused on the main submodules of speaker
diarization, namely, speaker embedding, segment similarity, and segment
clustering, where the VAD is left out of the scope of this study.

In the self-supervised learning (SSL) setting, unlabeled inputs define an
auxiliary task that can generate pseudo-labeled training data. Then, a model
can be trained using supervised learning techniques, and typically with the
contrastive loss function, which aims to learn a metric indicating that a pair
of positive and negative examples are dissimilar. For example, the SSL model
in [8] is designed to distinguish between an adjacent frame, which is
considered a positive example, and a randomly sampled future frame in the
sequence, which is considered a negative example.

In the supervised diarization setting, a positive and negative pair can
correspond to the same speaker and different speaker segments, respectively.
In the unsupervised setting, on the other hand, we have no supervision of who
spoke when, hence we cannot easily generate positive and negative pairs. We
cannot confidently assume that randomly sampled frames in an audio sequence
represent a different speaker and hence considered as the negative frames as
proposed in [8]. Additionally, we cannot sample negative segments from
different audio recordings as this can introduce other artifacts, and the
negative examples might represent different acoustic environments rather than
different speakers.

Due to these constraints, we take inspiration from recent developments in SSL
for image representation [9, 10] that do not require negative examples. In
those works, pairs of positive examples are generated by creating two
distortions of the same image. The objective of the learning algorithm is to
create a representation embedding invariant to distortions. In our setting,
the inputs to the proposed models are raw speech segments considered to be
spoken by the same speaker. Our setting aims to create a speaker-embedding
that is invariant to the segment spoken by the speaker, and it is achieved by
maximizing the agreement between embedding vectors of two different segments
of the same speaker (rather than two distortions of the same speech segment.)

The self-supervised speaker embeddings is then used to create a development
set of pseudo-labels, which, in turn, are used to train a PLDA-based
similarity matrix to the segments in a given audio stream. Additionally, the
pseudo-labels are used to tune the speaker clustering stopping threshold.

The contribution of this paper is a full unsupervised model for training a
speaker diarization system, including all its sub-components: a speaker
embedding model, a PLDA-based scoring matrix, and the detection thresholds.
The results show that our unsupervised method is better than other
unsupervised methods when two speakers are concerned, and comparable to the
state-of-the-art fully-supervised methods from just a couple years ago. Our
implementation is available at https://github.com/MLSpeech/ssl_diarization.

Related work.  There are few methods for unsupervised diarization that have
been previously proposed. Castaldo _et al._ [11] proposed an expectation-
maximization (EM) algorithm to the problem. First, a blind segmentation of the
audio stream is performed. Then, a speaker model is estimated on the current
segmentation to become the new global model or update the nearest speaker
global model. Shum _et al._ [12] also proposed an iterative algorithm for
speaker clustering based on principal component analysis and variational
inference. Finally, the work [13] proposes to use Bayesian Information
Criterion (BIC) for creating training data and training a PLDA similarity
metric on top of it. All those methods are based on i-vectors of GMM-UBM.
Furthermore, they all need annotated data for tuning thresholds or biases. In
contrast, our approach relies on recent advances in DNN-based speaker
representation and does not use any annotated data for tuning thresholds or
biases.

##  2 Model

Figure 1: An illustration of our embeddings model and training scheme.

We denote a speech utterance of $T$ samples as
$\mathbf{x}=\left(x_{1},\ldots,x_{T}\right)$, where
$\mathbf{x}\in\mathbb{R}^{T}$. We assume the existence of a label set
$\mathcal{Y}=\left\\{\phi,1,\ldots,K\right\\}$, where $\phi$ is the symbol for
silence or non-speech, and $K$ is the number of assumed speakers. In our
setting, the value of $K$ is unknown at training time or during the inference,
nor the labels associated with the inputs. We used the labels only for final
evaluation purposes.

We denote by $\mathbf{x}_{t}=\mathbf{x}_{t-T^{\prime}}^{t+T^{\prime}}$ an
audio segment of $2T^{\prime}$ samples around the $t$-th sample, and we denote
by $N$ the number of segments in the utterance. Our goal is to learn a
function that predicts the speaker value $y_{t}\in\mathcal{Y}$ for each
segment, where it is assumed there is only one speaker within each segment.
The proposed model is built from three components: an encoder $f_{\theta}$
with parameters $\theta$, a similarity scoring matrix $S\in\mathbb{R}^{N\times
N}$ between the $N$ segments of the utterance, and a hierarchical clustering
of the scoring matrix.

The encoder $f_{\theta}$ is depicted in Figure 1. It gets as input a speech
segment $\mathbf{x}_{t}$ and output an embedding vector
$\mathbf{z}_{t}=f_{\theta}(\mathbf{x}_{t})$. Ideally, the encoder should be
trained using a contrastive loss function. However, since we assume we do not
have access to segments of different speakers (negative examples), the encoder
is trained using a non-contrastive self-supervised loss function. Our loss
function gets two representations assumed to be of the same speaker as input.
We denote these representations by $\mathbf{z}_{t}$ and $\mathbf{z}_{\tau}$
($t\neq\tau$,) which corresponds to input segments $\mathbf{x}_{t}$ and
$\mathbf{x}_{\tau}$, respectively.

Denote the _cross-correlation matrix_ between the two embeddings as
$\mathbf{R}_{\mathbf{Z}_{t}\mathbf{Z}_{\tau}}$:

$\mathbf{R}_{\mathbf{Z}_{t}\mathbf{Z}_{\tau}}=\mathbb{E}\left[\mathbf{z}_{t}^{\top}\mathbf{z}_{\tau}\right],$
(1)

where $\mathbf{Z}_{t}$ and $\mathbf{Z}_{\tau}$ denote the random variable
associated with the vectors $\mathbf{z}_{t}$ and $\mathbf{z}_{\tau}$. Our
self-supervised loss function is aimed at making the cross-correlation matrix
$\mathbf{R}_{\mathbf{Z}_{t}\mathbf{Z}_{\tau}}$ as close as possible to the
identity matrix, $\mathbf{I}$, under the Frobenius norm:

$L_{BT}(\mathbf{z}_{t},\mathbf{z}_{\tau})=\|\mathbf{R}_{\mathbf{Z}_{t}\mathbf{Z}_{\tau}}-\mathbf{I}\|^{2}_{\mathcal{F}}~{}.$
(2)

By using this loss, the diagonal elements of the matrix are steered towards 1
to make the embeddings invariant to different segments of the same speaker. In
contrast, the off-diagonal elements are steered to 0, thus decorrelating the
vector components of the embeddings. This loss function is known as the
_Barlow Twins_ loss [9]. Practically, the cross-correlation matrix is computed
over the examples within the batch, and the parameters $\theta$ of the encoder
function $f_{\theta}$ are found by minimizing this loss function on the
unlabeled training set of utterances.

For subsequent processing, we generate a dataset with _pseudo-speaker labels_.
Our challenge is reliably labeling this dataset. First, we compute the
embeddings of the utterances in our _unlabeled development set_ using the
trained encoder. Then, we compute the Euclidean distances between these
embeddings to generate the distance matrix $\mathbf{D}$ whose elements are:

$[\mathbf{D}]_{i,j}=\|\mathbf{z}_{i}-\mathbf{z}_{j}\|^{2}_{2}~{}.$

Next, we cluster the matrix $\mathbf{D}$ using agglomerative hierarchical
clustering (AHC), resulting in a dendrogram, which is a diagram of the
hierarchical relationship between segments. We decide on a cutoff point in the
dendrogram that results in a number of clusters we assume to be higher than
the speaker count of any file in our data, which consequently enables us to
assume with high confidence cluster purity, i.e., the embeddings contained
within each cluster – all represent the same speaker. The higher the cutoff,
the higher the confidence. Then, we take the embeddings from the largest-sized
cluster from each utterance and label them as a pseudo-speaker. We repeat this
process for each utterance in the development set, using a new pseudo-speaker
label each time.

The second part of our system is a similarity scoring model $\mathbf{S}$,
which outputs the similarity scores $\mathbf{S}(\mathbf{z})_{i,j}$ between
$\mathbf{z}_{i}$ and $\mathbf{z}_{j}$ for all $i$ and $j$ segments of each
utterance $\mathbf{z}$. The scoring model $\mathbf{S}$ is obtained by training
a probabilistic linear discriminant analysis (PLDA) on the aforementioned
development set in the same fashion as in the supervised setting, but with the
pseudo generated labels [14].

Finally, given an encoded utterance $\mathbf{z}$, we segment $\mathbf{z}$ to
speakers by applying AHC on the similarity matrix $\mathbf{S}(\mathbf{z})$.
For this, we need to select the threshold for the stopping criteria of the
hierarchical clustering in an _unsupervised_ manner. We use our self-labeled
development set to randomly sample segments (10 frames) from each pseudo-
speaker (with repetitions). Next, we randomly concatenate these segments to
create utterances with multiple speakers. We compute the similarity matrix for
each generated utterance and use the $i$-th row of the similarity matrix
$\mathbf{S}(\mathbf{z})$, denoted $\mathbf{S}(\mathbf{z})_{i}$, to represent
the $i$-th segment of an input utterance. We set the similarity metric between
segment $i$ and segment $j$ as
$\|\mathbf{S}(\mathbf{z})_{i}-\mathbf{S}(\mathbf{z})_{j}\|^{2}$.
Traditionally, the PLDA scores themselves are used for this clustering step.
However, during our experiments, we found that using the row representations
results in a more robust threshold than when merely using the pre-computed
scores. Using these scores, we tune the clustering threshold on the
development dataset and select the threshold that yields the lowest
Diarization Error Rate (DER) on the pseudo labels. Empirically, we found that
the resulting threshold also generalizes well to the test data.

##  3 Empirical Evaluation

In this section we describe a set of experiments to demonstrate the efficiency
of the proposed method.

###  3.1 Datasets

For training, we used Fisher English Training Speech Part 1 (LDC2004S13) and
Part 2 (LDC2005S13), CallFriend, Switchboard-1 Release 2 [15], RuSTeN
(LDC2006S34), Gulf (LDC2006S43) and Iraq (LDC2006S45) Arabic Conversational
Telephone Speech. Our development and test sets were different parts of
CallHome. Supervised methods [16, 17, 18, 19, 20, 21] commonly reported
Diarization error rate (DER) results on a subset of the CallHome with a
development set consisting of 155 two-speaker calls, and a test set of 148
two-speaker calls prepared in [21]. For a fair comparison we use the same
subset in Table 2 and Table 3. Unsupervised methods [11, 12] reported DER on
the full multi-speaker test set of CallHome (the NIST SRE 2000 set), which we
also use in Table 1. Since the two-speaker development set is a subset of the
full Callhome test, in this setting we used a different subset of the NIST SRE
set, namely disks six and seven, where there are more than one speaker per
call, as the development set.

###  3.2 Experimental setting

We start with the speaker-discriminative embeddings training. The
aforementioned training datasets were all converted into single-channel
recordings. We applied VAD111We used Google’s WebRTC with sensitivity 1 and
frame size of 500 msec. It can be accessed at https://github.com/wiseman/py-
webrtcvad. to remove silences and non-speech. We randomly selected the inputs
$\mathbf{x}_{t}$ and $\mathbf{x}_{\tau}$, where our segment size was 500 msec,
and the segment pairs were 500 msec apart (using two adjacent segments
resulted in an overfitting which did not generalize to good speaker
separation). The encoder $f_{\theta}$ consists of 7 layers of 1-D convolutions
(kernel sizes: 10,10,10,8,4,4,4; strides: 5,5,5,4,2,2,2; 128 channels per
layer). Following this, there are 3 linear layers, each with 512 output units.
The first 2 layers are followed by batch normalization and ReLU. We used the
LARS optimizer [22]. We set a learning rate of 0.2 for the weights and 0.0048
for the biases and batch normalization parameters. We used a learning rate
warm-up period of 10 epochs, after which we reduced the learning rate using a
cosine decay schedule [23]. The model was trained using a batch size of 2048.
We trained the encoder $f_{\theta}$ until convergence (on our data, at around
30 epochs). This convergence also correlates well with an approximated DER we
computed by clustering the produced embeddings. The resulting embeddings
represent a 500 msec segment of speech.

To generate a self-supervised development set with pseudo labels, we took the
development subset of CallHome and extracted embeddings using our encoder.
Next, we clustered the embeddings using AHC. We cutoff the resulting
dendograms arbitrarily at 10 clusters, selected the largest cluster and
assumed cluster purity in the sense of it consisting of same speaker segments.
We assumed each resulting cluster from different files corresponded to
different speakers. To test our assumptions regarding the clustering purity,
we checked DER by labeling only the segments of the largest cluster as a
single speaker in each file and marked the rest of the file as silence. We got
only a 0.3% speaker error, along with an extremely high percentage of missed
speech which is to be expected. This validates our assumption that each of the
selected clusters consists almost entirely of a single speaker, creating
accurate pseudo-labels for a development dataset.

The PLDA model was then trained on this pseudo labeled development set in the
same fashion as in the supervised setting, using Kaldi’s setup [24]. We also
found a stopping threshold for the hierarchical clustering using this data by
randomly generating 20,000 utterances.

###  3.3 Results

Table 1: Speaker error rate on the multi-speaker test set of CallHome compared with unsupervised methods. #speakers | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $avg$
---|---|---|---|---|---|---|---
Castaldo [11] | $8.7$ | $15.7$ | $15.1$ | $20.2$ | $25.5$ | $29.8$ | $13.7$
Schum [12] | $14.5$ | ${\bf 14.0}$ | ${\bf 13.0}$ | ${\bf 17.0}$ | ${\bf 19.5}$ | ${\bf 24.0}$ | $14.5$
Ours | ${\bf 6.1}$ | $14.9$ | $18.3$ | $23.0$ | $30.7$ | $30.5$ | ${\bf 13.4}$
#calls | $303$ | $136$ | $43$ | $10$ | $6$ | $2$ | $500$
Table 2: Comparison of our unsupervised model with recent supervised methods. DER is reported on the two speaker test set of CallHome prepared in [21].  Model | DER
---|---
UIS-RNN V1 [25] | $10.6$
UIS-RNN V2 [25] | $9.6$
UIS-RNN V3 [25] | $7.6$
x-vector + LSTM (oracle VAD) [5] | $6.6$
DIVE [17] | $5.9$
Ours (unsupervised, oracle VAD) | 6.6
Ours (unsupervised) | 9.1

We turn to evaluate our method against state-of-the-art unsupervised and
supervised models. Then, we demonstrate the effect of the amount of
(unsupervised) data on the system’s performance. Finally, we conclude with a
detailed ablation study to gauge the impact of each component of our system.
All measurements were obtained using the NIST DER tool with a collar of 250
msec and ignore overlapping sections.

We start by presenting our work compared with other unsupervised works
mentioned in Section 1. In these papers, the full multi-speaker test set of
CallHome was used. This amounts to 500 recordings, each 2-5 minutes in length,
containing between 2 and 7 participants.

Table 1 reports the confusion error per speaker count and the average speaker
error rate using oracle VAD of our work compared with two unsupervised
methods: Castaldo _et al._ [11] and Shum _et al._ [12]. Since an oracle VAD is
employed in these measurements, we exclude false-alarms and miss-detection
from our evaluations, and report speaker confusion only. Our system is better
for the two speakers setting and our overall (weighted) results are better
than both methods.

Next, we would like to compare the performance of our unsupervised system to
the performance of state-of-the-art supervised systems from recent years. Note
that supervised systems are often evaluated on the two-speakers subset of the
dataset that used in the evaluation of unsupervised methods. Table 2 reports
DER on the two speakers test set of CallHome compared to recent state-of-the-
art _supervised_ methods. UIS-RNN [25] is a hybrid system training an RNN on
top of pre-trained speaker embeddings, with the three version: V1 was trained
on 36M utterances from 18K US English speakers (all mobile phone data); V2 was
trained on additional non-public containing 34M utterances from 138K speakers,
LibriSpeech, VoxCeleb, and VoxCeleb2, and V3 was based on variable-length
processing windows, drawn uniform at random from [240 msec, 1600 msec]. DIVE
[17] combines three modules which are trained jointly: projection of the
waveform to an embedding space, iterative selection of long-term speaker
representations, and per-speaker per-timestep VAD and is the current state of
the art reaching 5.9% without overlap. Similar to UIS-RNN, [5] suggested
training an LSTM to model the similarity between pretrained speaker embeddings
and then to perform the diarization. Unlike the other models in Table 2, it
uses oracle speech activity labels (removing silences).

The last rows are the result of our model, with learned embeddings, PLDA, and
clustering tuned on pseudo labels, with and without overlap. The error
breakdown without overlap is as follows: 1.5% missed-speech, 2.2% false-alarm,
and 5.2% speaker error. For a fair comparison, we added the performance of our
system against using the oracle VAD labels. In this setting, we achieve a
6.58% DER comparable to their 6.63%. As seen, our unsupervised system is only
slightly worse than state-of-the-art supervised systems from recent years.

Figure 2: Drop in DER as a function of the amount of hours of training data
used for training.

In Figure 2 we show that the more unlabeled data that is used, the better the
embedding model. Here we present different combinations of datasets we used
and the total hours of training data used in each mode. The total hours are
calculated by summing all the VAD segments longer than 3 seconds, as this is
what was used for training and the DER is the calculated by clusering the
embeddings using the number of speakers as a stopping criteria. As seen the
DER ranges from 28% for 50 hours of data to 11.5% for 2400 hours. This bodes
well for the system as most diarzation users will have access to large amounts
of in domain unlabeled data.

Table 3: An ablation study of the different system components. Results are reported on the two-speaker test set of CallHome. Model | DER
---|---
Embeddings + clustering + #spk | $11.5$
Embeddings + PLDA + clustering | $9.1$
Embeddings + PLDA + #spk | $8.7$
Embeddings + PLDA + clust. + (oracle VAD) | $6.6$
VicReg embeddings + clustering + #spk | $13.1$

Last, we evaluated our system by methodically omitting the different
components. The results in terms of DER [%] on the test set of CallHome are
reported in Table 3.

In the table, _Embeddings + clustering + #spk_ denotes the results when using
only the learned speaker-embeddings without incorporating the learned PLDA
similarity matrix, but where the speaker count is known and is used as a
stopping criterion for the clustering procedure. This is essentially a measure
of the goodness of the learned speaker embeddings. _Embeddings + PLDA +
clustering_ refers to our final choice of architecture, as described in
Section 2.

_Embeddings + PLDA + #spk_ is our final model when the speaker count in each
file is known and used as the clustering stopping criteria. As expected,
knowing the speaker count in each file yields slightly better results. These
results suggest that incorporating our proposed self-supervised PLDA
similarity matrix improved results by 24% in relative terms over pure
embeddings. _Embeddings + PLDA + oracle VAD_ denotes the results when using
the oracle VAD. This shows that improving the VAD can significantly lower the
DER.

For completeness, we also trained the speaker-embedding system using the
_VicReg_ objective proposed in [10]. This objective similar to _Barlow Twins_
, exchanges the variance term with a hinge loss function and adds a term
minimizing the MSE between embedding vectors. We denote this setting by
_VecReg embeddings + clustering + #spk_. However results suggest that this
system was inferior by approximately 1.5% absolute DER.

##  4 Discussion and future work

This paper introduces a new approach for completely unsupervised speaker
diarization. We showed that this method with a relatively small model,
performs close to state-of-the-art fully supervised methods on the CallHome
test set, further closing the gap between supervised and unsupervised methods.

For future work, we will explore ways of dealing with overlapping speech,
which is not referenced in this work. Additionally, we will examine ways to
incorporate negative examples in a way that can further improve results.
Finally, we are interested in studying how leveraging massive amounts of
unlabeled data affects this model.

## References

  * [1] N. Dehak, P. J. Kenny, R. Dehak, P. Dumouchel, and P. Ouellet, “Front-end factor analysis for speaker verification,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 19, no. 4, pp. 788–798, 2010.
  * [2] D. Snyder, D. Garcia-Romero, G. Sell, D. Povey, and S. Khudanpur, “X-vectors: Robust dnn embeddings for speaker recognition,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 5329–5333.
  * [3] L. Wan, Q. Wang, A. Papir, and I. L. Moreno, “Generalized end-to-end loss for speaker verification,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 4879–4883.
  * [4] S. Ioffe, “Probabilistic linear discriminant analysis,” in _European Conference on Computer Vision_. Springer, 2006, pp. 531–542.
  * [5] Q. Lin, R. Yin, M. Li, H. Bredin, and C. Barras, “Lstm based similarity measurement with spectral clustering for speaker diarization,” _arXiv preprint arXiv:1907.10393_ , 2019.
  * [6] W. H. Day and H. Edelsbrunner, “Efficient algorithms for agglomerative hierarchical clustering methods,” _Journal of classification_ , vol. 1, no. 1, pp. 7–24, 1984.
  * [7] T. Fuchs and J. Keshet, “Spoken term detection automatically adjusted for a given threshold,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 11, no. 8, pp. 1310–1317, 2017.
  * [8] F. Kreuk, J. Keshet, and Y. Adi, “Self-supervised contrastive learning for unsupervised phoneme segmentation,” _arXiv preprint arXiv:2007.13465_ , 2020\.
  * [9] J. Zbontar, L. Jing, I. Misra, Y. LeCun, and S. Deny, “Barlow twins: Self-supervised learning via redundancy reduction,” _arXiv preprint arXiv:2103.03230_ , 2021.
  * [10] A. Bardes, J. Ponce, and Y. LeCun, “Vicreg: Variance-invariance-covariance regularization for self-supervised learning,” _arXiv preprint arXiv:2105.04906_ , 2021.
  * [11] F. Castaldo, D. Colibro, E. Dalmasso, P. Laface, and C. Vair, “Stream-based speaker segmentation using speaker factors and eigenvoices,” in _2008 IEEE International Conference on Acoustics, Speech and Signal Processing_. IEEE, 2008, pp. 4133–4136.
  * [12] S. H. Shum, N. Dehak, R. Dehak, and J. R. Glass, “Unsupervised methods for speaker diarization: An integrated and iterative approach,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 21, no. 10, pp. 2015–2028, 2013.
  * [13] G. Le Lan, S. Meignier, D. Charlet, and P. Deléglise, “Speaker diarization with unsupervised training framework,” in _2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2016, pp. 5560–5564.
  * [14] J. Prazak and J. Silovsky, “Speaker diarization using plda-based speaker clustering,” in _Proceedings of the 6th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems_ , vol. 1. IEEE, 2011, pp. 347–350.
  * [15] J. J. Godfrey, E. C. Holliman, and J. McDaniel, “Switchboard: Telephone speech corpus for research and development,” in _Acoustics, Speech, and Signal Processing, IEEE International Conference on_ , vol. 1. IEEE Computer Society, 1992, pp. 517–520.
  * [16] S. Horiguchi, Y. Fujita, S. Watanabe, Y. Xue, and K. Nagamatsu, “End-to-end speaker diarization for an unknown number of speakers with encoder-decoder based attractors,” _arXiv preprint arXiv:2005.09921_ , 2020.
  * [17] N. Zeghidour, O. Teboul, and D. Grangier, “Dive: End-to-end speech diarization via iterative speaker embedding,” _arXiv preprint arXiv:2105.13802_ , 2021\.
  * [18] Y. Fujita, N. Kanda, S. Horiguchi, K. Nagamatsu, and S. Watanabe, “End-to-end neural speaker diarization with permutation-free objectives,” _arXiv preprint arXiv:1909.05952_ , 2019.
  * [19] Y. Fujita, S. Watanabe, S. Horiguchi, Y. Xue, and K. Nagamatsu, “End-to-end neural diarization: Reformulating speaker diarization as simple multi-label classification,” _arXiv preprint arXiv:2003.02966_ , 2020.
  * [20] S. Horiguchi, P. García, Y. Fujita, S. Watanabe, and K. Nagamatsu, “End-to-end speaker diarization as post-processing,” in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 7188–7192.
  * [21] Y. Fujita, N. Kanda, S. Horiguchi, Y. Xue, K. Nagamatsu, and S. Watanabe, “End-to-end neural speaker diarization with self-attention,” in _2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_. IEEE, 2019, pp. 296–303.
  * [22] Y. You, I. Gitman, and B. Ginsburg, “Large batch training of convolutional networks,” _arXiv preprint arXiv:1708.03888_ , 2017.
  * [23] I. Loshchilov and F. Hutter, “Sgdr: Stochastic gradient descent with warm restarts,” _arXiv preprint arXiv:1608.03983_ , 2016.
  * [24] D. Povey, A. Ghoshal, G. Boulianne, L. Burget, O. Glembek, N. Goel, M. Hannemann, P. Motlicek, Y. Qian, P. Schwarz _et al._ , “The kaldi speech recognition toolkit,” in _IEEE 2011 workshop on automatic speech recognition and understanding_ , no. CONF. IEEE Signal Processing Society, 2011.
  * [25] A. Zhang, Q. Wang, Z. Zhu, J. Paisley, and C. Wang, “Fully supervised speaker diarization,” in _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 6301–6305.


# Histogram Layer Time Delay Neural Networks for Passive Sonar Classification

###### Abstract

Underwater acoustic target detection in remote marine sensing operations is
challenging due to complex sound wave propagation. Despite the availability of
reliable sonar systems, target recognition remains a difficult problem.
Various methods address improved target recognition. However, most struggle to
disentangle the high-dimensional, non-linear patterns in the observed target
recordings. In this work, a novel method combines a time delay neural network
and histogram layer to incorporate statistical contexts for improved feature
learning and underwater acoustic target classification. The proposed method
outperforms the baseline model, demonstrating the utility in incorporating
statistical contexts for passive sonar target recognition. The code for this
work is publicly available.

Index Terms—  Deep learning, histograms, passive sonar, target classification,
texture analysis

##  1 Introduction

Figure 1: Overall experimental work flow. Each signal is resampled to $16$ kHz
and binned into three second segments. After dividing the signals and
corresponding segments into training, validation, and test partitions, several
time-frequency features are extracted. The features are then passed into the
model and classified as one of the four vessel types.

Underwater acoustic target recognition (UATR) technology plays a crucial role
in a variety of domains, including biology [1], carrying out search and rescue
operations, enhancing port security [2], and mapping the ocean floor [3]. One
of the primary target detection techniques used by modern crafts, such as
unmanned underwater vehicles, is passive sonar [4]. Passive sonar is an
underwater acoustic technology that uses hydrophones to detect and analyze
sound waves in the ocean [5]. Unlike active sonar, passive sonar resolves
targets from the natural sounds of the ocean and the noises produced by ships
and other underwater vehicles. Processing and analyzing passive sonar data can
be challenging due to the high volume of data and environmental complexity
[6]. Signal processing techniques are often used to analyze ship-generated
noise such as low frequency analysis and recording (LOFAR) spectra [7]. The
Detection of Envelope Modulation on Noise (DEMON) is an approach that has been
successfully used for target detection and recognition in passive sonar [8, 9,
10]. Despite their success, these approaches use handcrafted features that can
be difficult to extract without domain expertise [11].

Figure 2: Proposed HLTDNN architecture. The histogram layer is added in the
parallel with the baseline TDNN model through the bin center and width
convolution layers with the radial basis activation function (RBF) and average
pooling layer.

Artificial neural networks (ANNs), such as convolutional neural networks
(CNNs) and time delay neural networks (TDNNs), provide an end-to-end process
for automated feature learning and follow-on tasks (e.g., detection and
classification of signals) [12, 13, 14, 15]. The TDNN has shown success in
simulating long-term temporal dependencies [16] and can be modeled as a 1D CNN
[13]. Thus, the TDNN can adaptively learn the sequential hierarchies of
features, but does not explicitly account for the statistics of passive sonar
data. These are difficult to model for feature extraction [17, 18]. The
statistics of the signals can describe the acoustic texture of the targets of
interest [18]. Texture generally falls into two categories: statistical and
structural [19, 20, 21, 22].Statistical context in audio analysis involves
studying the amplitude information of the audio signal. One way to capture
amplitude information is by using probability density functions [18]. However,
traditional artificial neural network (ANN) approaches, like convolutional
neural networks (CNNs) and time-delay neural networks (TDNNs), have shown a
bias towards capturing structural textures rather than statistical texture
[20, 21, 22]. This bias limits their ability to directly model the statistical
information required to capture acoustic textures accurately. To overcome this
shortcoming, histogram layers can be integrated into ANNs to incorporate
statistical context [22]. Methods that combine both structural and statistical
textures have improved performance for other tasks such as image
classification and segmentation [20, 21, 22]. In this work, we propose a new
TDNN architecture that integrates histogram layers for improved target
classification. Our proposed workflow is summarized in Figure 1. The
contributions of this work are as follows:

  * •

Novel TDNN architecture with histogram layer (HLTDNN) for passive sonar target
classification

  * •

In-depth qualitative and quantitative comparisons of TDNN and HLTDNN across a
suite of time-frequency features.

##  2 Method

###  2.1 Baseline TDNN Architecture

The TDNN architecture consisted of several convolution layers with the ReLU
activation function and max pooling. 2D convolutional features were extracted
from the time-frequency input to capture local relationships between the
vessel’s frequency information [23]. Padding was added to the input time-
frequency feature to maintain the spatial dimensions of the resulting features
maps. After each convolution operation and ReLU activation function, the
features were pooled along the time axis with desired kernel length $L$ (e.g.,
max pooling kernel of size $1\times L$) to aggregate the feature information
while maintaining the temporal dependencies similar to other TDNNs [16, 23].
After the fourth convolutional block, the features are flattened and then
passed through a final 1D convolutional layer followed by a sigmoid activation
function and global average pooling layer (GAP).

###  2.2 Proposed HLTDNN

The baseline TDNN is focused on the “structural” (e.g., local) acoustic
textures of time and frequency as well as the temporal dependencies in the
data. However, the model does not directly consider the statistical aspects of
the data. A histogram layer [22] can be added in parallel to the baseline TDNN
model to capture statistical features to assist in improving classification
performance. Given input features, $\mathbf{X}\in\mathbb{R}^{M\times N\times
D}$, where $M$ and $N$ are the spatial (or time-frequency) dimensions while
$D$ is the feature dimensionality, the output tensor of the local histogram
layer with $B$ bins, $\mathbf{Y}\in\mathbb{R}^{R\times C\times B\times D}$
with spatial dimensions $R$ and $C$ after applying a histogram layer with
kernel size $S\times T$ is shown in (1):

$Y_{rcbd}=\cfrac{1}{ST}\sum_{s=1}^{S}\sum_{t=1}^{T}e^{-\gamma_{bd}^{2}\left(x_{r+s,c+t,d}-\mu_{bd}\right)^{2}}$
(1)

where the bin centers ($\mu_{bd}$) and bin widths ($\gamma_{bd}$) of the
histogram layer are learnable parameters. Each input feature dimension is
treated independently, resulting in $BD$ output histogram feature maps. The
histogram layer takes input features and outputs the “vote” for a value in the
range of $[0,1]$. The histogram layer can be modeled using convolution and
average pooling layers as shown in Figure 2. Following previous work [22], the
histogram layer is added after the fourth convolutional block (i.e.,
convolution, ReLU, and max pooling) and its features are concatenated with the
TDNN features before the final output layer.

##  3 Experimental Procedure

###  3.1 Dataset Description

The DeepShip dataset [14] was used in this work. The database contained 609
records reflecting the sounds of four different ship types: cargo,
passengership, tanker, and tug. Following [14], each signal is re-sampled to a
frequency of $16$ kHz and divided into segments of three seconds. Figure 3
illustrates the structure of the dataset after “binning” the signals into
segments. The number of signals and segments for each class are also shown.

Figure 3: DeepShip dataset structure.

###  3.2 Experimental Design

Table 1: Overall performance metrics for baseline TDNN and proposed HLTDNN model. The average score with $\pm 1\sigma$ across the three experimental runs of random initialization is shown and the best average metric is bolded. The log of the Fisher Discriminant Ratio (FDR) is shown due to the magnitude of the FDR score. The time-frequency features in this work were Mel Spectrogram (MS), Mel-frequency cepstral coefficients (MFCC), Short-time Fourier transform (STFT), Gammatone-frequency cepstral coefficients (GFCC), Constant-q transform (CQT), and Variable-q transform (VQT). Features | Model | Accuracy | Precision | Recall | F1 Score | MCC | FDR
---|---|---|---|---|---|---|---
MS | TDNN |  50.31 $\pm$ 1.41% |  39.56 $\pm$ 0.05% |  47.67 $\pm$ 0.03% |  42.09 $\pm$ 0.02% |  34.22 $\pm$ 0.02% |  4.14 $\pm$ 1.50
HLTDNN |  47.46 $\pm$ 2.39% |  45.25 $\pm$ 0.03% |  51.80 $\pm$ 0.04% |  46.00 $\pm$ 0.03% |  29.55 $\pm$ 0.03% | 20.51 $\pm$ 1.86
MFCC | TDNN |  51.39 $\pm$ 0.79% |  50.10 $\pm$ 0.02% |  49.95 $\pm$ 0.03% |  49.48 $\pm$ 0.02% |  34.84 $\pm$ 0.01% |  5.34 $\pm$ 1.29
HLTDNN |  54.41 $\pm$ 0.42% |  54.28 $\pm$ 0.03% |  53.91 $\pm$ 0.03% | 53.62 $\pm$ 0.02% |  39.38 $\pm$ 0.02% |  15.29 $\pm$ 1.85
STFT | TDNN |  51.15 $\pm$ 0.72% |  40.88 $\pm$ 0.03% |  48.49 $\pm$ 0.01% |  43.86$\pm$ 0.02% |  24.04 $\pm$ 0.04% |  8.30 $\pm$ 2.87
HLTDNN | 59.21 $\pm$ 0.56% | 54.84 $\pm$ 0.02% | 56.59 $\pm$ 0.03% |  53.23 $\pm$ 0.02% | 46.05 $\pm$ 0.01% |  17.75 $\pm$ 0.58
GFCC | TDNN |  27.73 $\pm$ 0.18% |  17.45 $\pm$ 0.00% |  26.40 $\pm$ 0.00% |  17.61 $\pm$ 0.00% |  3.63 $\pm$ 0.00% |  15.26 $\pm$ 0.44
HLTDNN |  43.42 $\pm$ 0.61% |  39.63 $\pm$ 0.01% |  41.44 $\pm$ 0.01% |  38.57 $\pm$ 0.01% |  24.24 $\pm$ 0.01% |  11.94 $\pm$ 4.82
CQT | TDNN |  36.89 $\pm$ 0.83% |  23.34 $\pm$ 0.03% |  34.92 $\pm$ 0.07% |  30.85 $\pm$ 0.02% |  15.06 $\pm$ 0.01% |  16.95 $\pm$ 0.56
HLTDNN |  50.66 $\pm$ 1.37% |  44.37 $\pm$ 0.01% |  48.04 $\pm$ 0.02% |  43.62 $\pm$ 0.02% |  34.30 $\pm$ 0.02% |  13.14 $\pm$ 3.61
VQT | TDNN |  36.76 $\pm$ 0.96% |  28.14 $\pm$ 0.02% |  34.80 $\pm$ 0.07% |  30.76 $\pm$ 0.02% |  14.84 $\pm$ 0.01% |  16.82 $\pm$ 0.94
HLTDNN |  50.12 $\pm$ 0.27% |  43.35 $\pm$ 0.02% |  47.57 $\pm$ 0.01% |  43.40 $\pm$ 0.01% |  33.44 $\pm$ 0.00% |  13.28 $\pm$ 2.87

Feature Extraction Six different features are extracted: Mel Spectrogram (MS),
Mel-frequency cepstral coefficients (MFCCs), Short-time Fourier transform
(STFT), Gammatone-frequency cepstral coefficients (GFCC), Constant-q transform
(CQT), and Variable-q transform (VQT). The window and hop length for each
feature was set to $250$ and $64$ ms respectively [14]. The number of Mel
filter banks for the MelSpectrogram was set to $40$. For MFCC, the number of
Mel-frequency ceptral coeffiencts was $16$. The number of frequency bins for
STFT was $48$ while GFCC, CQT, and VQT used 64 frequency bins. The feature
dimensions after zero-padding were $48\times 48$ for MS and STFT, $16\times
48$ for MFCC, and $64\times 48$ for GFCC, CQT, and VQT.

Data partitioning The data set was split into 70% training, 15% validation,
and 15% test based on the signals (428 training, 90 validation, and 91 test).
After “binning” the signals into three second segments, 56,468 segments were
created (38,523 training, 9,065 validation, and 8,880 test). All segments of
each signal remained in the same partition to prevent data leakage (i.e., if
one signal was selected for training, all segments of the signal were also
used for training).

Experimental setup The models (TDNN or HLTDNN) were evaluated with each
individual feature across three runs of random initialization. The
experimental parameters for the models were the following:

  * •

Optimizer: Adagrad

  * •

Learning rate ($\eta$): 0.001

  * •

Batch size: 128

  * •

Epochs: 100

  * •

Dropout ($p$): 0.5

  * •

Early stopping: 10 epochs

  * •

Number of bins (HLTDNN): 16

Dropout was added before the output classification layer and early stopping
was used to terminate training if validation loss did not improve within
number of patience epochs. Experiments were conducted on an NVIDIA RTX 3090.
The models are implemented in Pytorch 1.13, TorchAudio 2.0, and nnAudio 0.3.1
[24].

##  4 Results and Discussion

###  4.1 Classification Performance

(a) TDNN ($51.15\pm 0.72\%$)

(b) HLTDNN ($59.21\pm 0.56\%$)

Figure 4: Average confusion matrices for the TDNN and HLTDNN on the DeepShip
dataset using the STFT feature. The average overall test accuracy is shown in
parenthesis.

TDNN and HLTDNN classification performances are shown in Table 1.
Classification performance was accessed using five metrics: accuracy,
precision, recall, F1 score, and Matthew’s correlation coefficient (MCC).
Fisher’s discriminant ratio (FDR) was used to access the feature quality
(discussed more in Section 4.2). Confusion matrices for the TDNN and HLTDNN
using best performing feature are displayed in Figures 4(a) and 4(b)
respectively. For the HLTDNN, STFT achieved the best classification
performance compared to other features. However, MFCC had the best for
performance for TDNN across the different performance metrics. STFT performed
similarly to MFCC when observing classification accuracy. Additional
quantitative and qualitative analysis will use STFT to evaluate the impact of
the histogram layer on the vessel classification.

The TDNN model initially performed well with the Mel spectrogram, MFCC, and
STFT, but significantly degraded for the other three features (Table 1). The
best performance was achieved using the MFCC feature as input while the worst
feature was GFCC. A possible reason for this is that each feature used a 250
ms window and hop length of 64 ms. The short time frame may be limiting the
frequency domain and selecting the best frequency band greatly impacts
performance [25]. However, the performance of the HLTDNN was fairly robust
across the different time-frequency features. The STFT feature performed the
best for this model, and the HLTDNN also improved the performance of the GFCC,
CQT and VQT features significantly in comparison to the TDNN. This
demonstrates that the statistical context captured by the histogram layer is
useful for improving target classification.

Both models did not identify the Cargo class as well as the other vessel types
as shown in Figure 4. Particularly, the most common classification mistakes
occurred when the model predicted Cargo as Tanker (i.e., false positive).
Intuitively, this classification error makes sense because Tanker is a type of
Cargo ship (e.g., oil tanker [26]) and the sound produced by each ship maybe
similar. Also, the Cargo class in the DeepShip data has been noted to have
high intra-class variance [27]. As a result, the Cargo class was the most
difficult to classify. Feature regularization methods (i.e., constrastive
learning) can be incorporated into the objective function to mitigate intra-
class variance.

###  4.2 Feature Evaluation

Table 2: STFT Fisher’s discriminant ratio (FDR) scores for each class and overall. The average score with $\pm 1\sigma$ across the three experimental runs of random initialization is shown and the best average metric is bolded. The log of the FDR is shown due to the magnitude of the FDR score. The higher FDR score indicates better separability and compactness of the features in higher dimensional space for each class. Class | TDNN | HLTDNN
---|---|---
Cargo |  6.00$\pm$4.11 | 23.65$\pm$7.42
Passengership |  6.36$\pm$3.01 | 19.44$\pm$ 2.56
Tanker |  5.08$\pm$5.75 | 19.67$\pm$3.11
Tug |  13.01$\pm$1.89 | 20.69$\pm$ 3.08
Overall |  8.30$\pm$2.87 | 17.75$\pm$ 0.58

(a) TDNN ($8.30\pm 2.87$)

(b) HLTDNN ($17.75\pm 0.58$)

Figure 5: 2D t-SNE projections of features from penultimate layer of the best
performing TDNN and HLTDNN on the DeepShip dataset using the STFT feature.
Each t-SNE projection used the same initialization for a fair qualitative
comparison between the features of each model. The average overall log FDR is
shown in parenthesis. The higher FDR score indicates better separability and
compactness of the features in higher dimensional space.

In addition to the classification metrics, quality of the features was
accessed using Fisher’s Discriminant Ratio (FDR). FDR is the ratio of the
inter-class separability and the intra-class compactness. Ideally, the inter-
class separability should be maximized (i.e., different vessel types should be
“far away” from one another or have large distances between the classes in the
feature space) and the intra-class compactness should be minimized (i.e.,
samples from the same class should be “close” or have small distances between
one another in the feature space). As a result, the FDR should be maximized.
From Table 1, the log of the FDR shows that the histogram model achieved the
best FDR scores for all six features further demonstrating the utility of the
statistical features.

A deeper analysis using the best performing feature (STFT) in terms of
classification performance is shown in Table 2. For all four classes, the log
FDR for the HLTDNN is statistically significant (no overlapping error bars) in
comparison to the TDNN. The main difference between the two models were the
increased feature separability of the HLTDNN model in comparison with the
baseline TDNN. The TDNN had smaller denominator (i.e., intra-class
compactness) compared to the HLTDNN when computing the norm of the within-
scatter matrix, indicating that the TDNN performs marginally better in terms
of intra-class compactness. On the other hand, the features from the HLTDNN
are more separable than those from the TDNN, as evident from the norm of the
between-scatter matrix, showing the HLTDNN’s superiority in terms of inter-
class separability. The FDR scores further elucidate the importance of
statistical texture information captured by the histogram layer.

Figure 5 shows the 2D t-SNE projection of the features from the best
performing models using the STFT feature. The same random initialization for
t-SNE was used for both methods in order to do a fair comparison between both
models. The qualitative results of t-SNE match our quantitative analysis using
FDR. The features extracted by the histogram acts as a similarity measure for
the statistics of the data and assigning higher “votes” to bins where features
are closer. The addition of these features to the TDNN model improved the
separability of the classes as observed in Figure 5(b). Modifying the
histogram layer to help improve the intra-class compactness of the HLTDNN
would be of interest in future investigations.

##  5 Conclusion

In this work, a novel HLTDNN model was developed to incorporate statistical
information for improved target classification in passive sonar. In comparison
to the base TDNN, the HLTDNN not only improved classification performance and
led to improved feature representations for the vessel types. Future work will
investigate combining features as opposed to using a single time-frequency
representation as the input to the network. Each feature can also be tuned
(e.g., change number of frequency bins) to enhance the representation of the
signals. Additionally, both architectures can be improved by a) adding more
depth and b) leveraging pretrained models. The training strategies could also
use approaches to mitigate overfitting and improve performance, such as
regularization of the histogram layer (e.g., add constraints to the bin
centers and widths) and data augmentation.

## References

  * [1] B. Beckler, A. Pfau, M. Orescanin, S. Atchley, N. Villemez, J. E. Joseph, C. W. Miller, and T. Margolina, “Multilabel classification of heterogeneous underwater soundscapes with bayesian deep learning,” _IEEE Journal of Oceanic Engineering_ , vol. 47, no. 4, pp. 1143–1154, 2022.
  * [2] K. R. Kita, S. Randeni, D. DiBiaso, and H. Schmidt, “Passive acoustic tracking of an unmanned underwater vehicle using bearing-doppler-speed measurements,” _The Journal of the Acoustical Society of America_ , vol. 151, no. 2, pp. 1311–1324, 2022.
  * [3] G. R. Mellema, “Reverse-time tracking to enhance passive sonar,” in _2006 9th International Conference on Information Fusion_. IEEE, 2006, pp. 1–8.
  * [4] Z. Cheng, X. Fan, L. Guo, and Y. Cui, “A UUV target detection method based on informer,” in _2022 4th International Conference on Frontiers Technology of Information and Computer (ICFTIC)_. IEEE, 2022, pp. 774–778.
  * [5] M. J. de Souza, N. N. de Moura Júnior, and J. M. de Seixas, “Passive sonar classification using time-domain information and recurrent neural networks,” in _2022 IEEE Latin American Conference on Computational Intelligence (LA-CCI)_. IEEE, 2022, pp. 1–6.
  * [6] D. Neupane and J. Seok, “A review on deep learning-based approaches for automatic sonar target recognition,” _Electronics_ , vol. 9, no. 11, p. 1972, 2020.
  * [7] X. Luo, L. Chen, H. Zhou, and H. Cao, “A survey of underwater acoustic target recognition methods based on machine learning,” _Journal of Marine Science and Engineering_ , vol. 11, no. 2, p. 384, 2023.
  * [8] L. Sichun and Y. Desen, “DEMON feature extraction of acoustic vector signal based on 3/2-d spectrum,” in _2007 2nd IEEE Conference on Industrial Electronics and Applications_ , 2007, pp. 2239–2243.
  * [9] M. A. R. Hashmi and R. H. Raza, “Novel DEMON spectra analysis techniques and empirical knowledge based reference criterion for acoustic signal classification,” _Journal of Electrical Engineering & Technology_, vol. 18, no. 1, pp. 561–578, 2023.
  * [10] S. K. Ambat _et al._ , “Performance evaluation of the DEMON processor for sonar,” in _2022 IEEE Region 10 Symposium (TENSYMP)_. IEEE, 2022, pp. 1–6.
  * [11] X. Cao, X. Zhang, Y. Yu, and L. Niu, “Deep learning-based recognition of underwater target,” in _2016 IEEE International Conference on Digital Signal Processing (DSP)_. IEEE, 2016, pp. 89–93.
  * [12] Y. Jing _et al._ , “A multilayered ANN architecture for underwater target tracking,” in _1994 Proceedings of Canadian Conference on Electrical and Computer Engineering_. IEEE, 1994, pp. 785–788.
  * [13] P. Ashok and B. Latha, “An improving recognition accuracy of underwater acoustic targets based on gated recurrent unit (GRU) neural network method,” in _2022 1st International Conference on Computational Science and Technology (ICCST)_. IEEE, 2022, pp. 1–6.
  * [14] M. Irfan, Z. Jiangbin, S. Ali, M. Iqbal, Z. Masood, and U. Hamid, “DeepShip: An underwater acoustic benchmark dataset and a separable convolution based autoencoder for classification,” _Expert Systems with Applications_ , vol. 183, p. 115270, 2021.
  * [15] V.-S. Doan, T. Huynh-The, and D.-S. Kim, “Underwater acoustic target classification based on dense convolutional neural network,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 19, pp. 1–5, 2020.
  * [16] V. Peddinti, D. Povey, and S. Khudanpur, “A time delay neural network architecture for efficient modeling of long temporal contexts,” in _Sixteenth Annual Conference of the International Speech Communication Association_ , 2015.
  * [17] H. Komari Alaie and H. Farsi, “Passive sonar target detection using statistical classifier and adaptive threshold,” _Applied Sciences_ , vol. 8, no. 1, p. 61, 2018.
  * [18] M. Trevorrow, “Examination of statistics and modulation of underwater acoustic ship signatures,” 2021.
  * [19] G. Srinivasan and G. Shobha, “Statistical texture analysis,” in _Proceedings of world academy of science, engineering and technology_ , vol. 36, no. December, 2008, pp. 1264–1269.
  * [20] D. Ji, H. Wang, M. Tao, J. Huang, X.-S. Hua, and H. Lu, “Structural and statistical texture knowledge distillation for semantic segmentation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 16 876–16 885.
  * [21] L. Zhu, D. Ji, S. Zhu, W. Gan, W. Wu, and J. Yan, “Learning statistical texture for semantic segmentation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 12 537–12 546.
  * [22] J. Peeples, W. Xu, and A. Zare, “Histogram layers for texture analysis,” _IEEE Transactions on Artificial Intelligence_ , vol. 3, no. 4, pp. 541–552, 2022.
  * [23] J. Thienpondt, B. Desplanques, and K. Demuynck, “Integrating frequency translational invariance in TDNNs and frequency positional information in 2d ResNets to enhance speaker verification,” in _Interspeech2021_. ISCA, 2021, pp. 2302–2306.
  * [24] K. W. Cheuk, H. Anderson, K. Agres, and D. Herremans, “nnAudio: An on-the-Fly GPU audio to spectrogram conversion toolbox using 1d convolutional neural networks,” _IEEE Access_ , vol. 8, pp. 161 981–162 003, 2020.
  * [25] A. Pollara, A. Sutin, and H. Salloum, “Improvement of the detection of envelope modulation on noise (DEMON) and its application to small boats,” in _OCEANS 2016 MTS/IEEE Monterey_. IEEE, 2016, pp. 1–10.
  * [26] C. Wang, H. Zhang, F. Wu, S. Jiang, B. Zhang, and Y. Tang, “A novel hierarchical ship classifier for COSMO-SkyMed SAR data,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 11, no. 2, pp. 484–488, 2013.
  * [27] L. Nie, C. Li, H. Wang, J. Wang, Y. Zhang, F. Yin, F. Marzani, and A. Bozorg Grayeli, “A contrastive-learning-based method for the few-shot identification of ship-radiated noises,” _Journal of Marine Science and Engineering_ , vol. 11, no. 4, p. 782, 2023.


# Paramagnetic spin correlations in colossal magnetoresistive La0.7Ca0.3MnO3

Joel S. Helton1,∗ Matthew B. Stone2 Dmitry A. Shulyatev3 Yakov M. Mukovskii3
Jeffrey W. Lynn1,† 1NIST Center for Neutron Research, National Institute of
Standards and Technology, Gaithersburg, Maryland 20899, USA  2Quantum
Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  3National University of Science and Technology “MISiS,” Moscow
119991, Russia

###### Abstract

Neutron spectroscopy measurements reveal dynamic spin correlations throughout
the Brillouin zone in the colossal magnetoresistive material La0.7Ca0.3MnO3 at
265 K ($\approx$1.03 $T_{C}$). The long-wavelength behavior is consistent with
spin diffusion, yet an additional and unexpected component of the scattering
is also observed in low-energy constant-$E$ measurements, which takes the form
of ridges of strong quasielastic scattering running along ($H$ 0 0) and
equivalent directions. Well-defined $Q$-space correlations are observed in
constant-$E$ scans at energies up to at least 28 meV, suggesting robust short-
range spin correlations in the paramagnetic phase.

###### pacs:

75.47.Gk, 75.40.Gb, 75.50.Cc, 78.70.Nx

Hole-doped perovskite manganites of the form La1-xCaxMnO3 (LCMO) feature a
ferromagnetic metallic ground state for 0.2 $<$ $x$ $<$ 0.5, with the highest
$T_{C}$ for the combined ferromagnetic and metal-insulator transition at an
optimal doping of $x$ $\approx$ 3/8.Wollan and Koehler (1955); Schiffer et al.
(1995); Cheong and Chen (1998) The colossal magnetoresistance (CMR) observed
at this transitionRamirez (1997) cannot be explained solely by Zener double-
exchange;Millis et al. (1995) rather, the physics underlying the CMR effect in
LCMO likely arises from strong coupling between the spin, charge, lattice, and
orbital degrees of freedomRöder et al. (1996); Varma (1996) and nanoscale
inhomogeneities between competing phases.Dagotto (2003, 2005) For $x$ = 0.3
LCMO, short-range static and dynamic polaron correlations are observed above
$T_{C}$Lynn et al. (2007); Bridges et al. (2010) with an ordering wave vector
of (0.25 0.25 0) signifying CE-typeGoodenough (1955) charge- and orbital-
ordered regions. The spin dynamics in the ferromagnetic phase are likewise
unconventional.Lynn (2000); Zhang et al. (2007) The spin wave dispersion
softens near the zone boundary, which can be fit to a phenomenological model
of first- and fourth-nearest-neighbor ferromagnetic Heisenberg interactions,
and displays anomalous spin wave damping.Ye et al. (2007); Dai et al. (2000)
The spin wave stiffness coefficient renormalizes but does not fully collapse
as $T_{C}$ is approached from below;Lynn et al. (1996) this contrasts with
higher-bandwidth manganite materials such as Pr1-xSrxMnO3 where the stiffness
fully collapses at $T_{C}$ as expected for a second-order ferromagnetic phase
transition.Fernandez-Baca et al. (1998) Above $T$ $\approx$ 200 K a spin
diffusive quasielastic component develops in the low-$q$ spectral weightLynn
et al. (1996); Dai et al. (2001) arising from a short-ranged localization of
electrons on the Mn3+/Mn4+ lattice. This quasielastic component displays a
strong field dependenceLynn et al. (1997) and dominates the spin fluctuation
spectrum near $T_{C}$ with a temperature dependence that closely matches those
of both lattice polarons and the bulk resistivity.Adams et al. (2000) It is
the development of this spin diffusive component, rather than the thermal
population of spin waves, that truncates the ferromagnetic metallic ground
state in a weakly first-order phase transition.Adams et al. (2004)

In this work we report neutron spectroscopy measurements on a single crystal
sample of La0.7Ca0.3MnO3. Spin correlations at 265 K, slightly above $T_{C}$,
are explored primarily through the $Q$ dependence of scattering at constant
energy transfer. The long-wavelength spin dynamics can be described by spin
diffusion with a short, almost temperature-independent correlation length of
$\approx$12 Å.Lynn et al. (1996) For $q$ transfers approaching the Brillouin
zone edge an additional anisotropic scattering component is observed at low
energies, in the form of ridges of strong quasielastic scattering intensity
running along ($H$ 0 0) and symmetry equivalent directions. To the best of our
knowledge, a component of this sort has not been reported in the paramagnetic
phase of any other isotropic ferromagnet. A close connection between the
low-$q$ quasielastic scattering and the colossal magnetoresistance has already
been established,Adams et al. (2000) supporting the possibility that the novel
high-$q$ paramagnetic scattering component presented here might also play a
role in the CMR physics. In the zone boundary ($H$ $K$ 0.5) plane spin
correlations are observed up to energies of at least 28 meV; this scattering
is qualitatively similar to the low-energy correlations in the ($H$ $K$ 0)
plane.

A 1.5 g single crystal sample of La0.7Ca0.3MnO3, previously used for
measurements of polaron correlations,Lynn et al. (2007) was grown by the
floating zone technique.Shulyatev et al. (2002) This is the highest
composition of LCMO for which single crystal samples have been successfully
grown, with $T_{C}$ = 257 K. The crystal structure is orthorhombic perovskite,
but given the presence of multiple crystallographic domains we employ the
cubic notation with $a$ = 3.87 Å. The experiment was carried out at the ARCS
time-of-flight chopper spectrometer at the Spallation Neutron Source, Oak
Ridge National Laboratory with an incident neutron energy of 50 meV at
temperatures of 100 K and 265 K. The 100 K spin wave dispersion in the ($\xi$
0 0) direction can be fit to $\hbar\omega$ =
2$S$[$|J_{1}|$(1-cos(2$\pi\xi$))+$|J_{4}|$(1-cos(4$\pi\xi$))] where the first-
and fourth-nearest neighbor ferromagnetic interactions are given by $S|J_{1}|$
= 6.18 $\pm$ 0.17 meV and $J_{4}/J_{1}$ = 0.19 $\pm$ 0.02, roughly in
agreement with the interaction values previously reportedYe et al. (2007) for
an LCMO sample with $T_{C}$ = 238 K.

Figure 1: (Color online) Intensity plot of $S(Q,\,\omega)$ measured at 265 K,
with the energy integrated between 3 meV $\leq$ $\hbar\omega$ $\leq$ 5 meV.
(a) Scattering in the ($H$ $K$ 0) plane. (b) Scattering in the (0 $K$ $L$)
plane. $Q$ perpendicular to the scattering plane has been integrated over
$\pm$ 0.16 Å-1. The black, red, purple, and orange dashed lines represent the
scan directions displayed in Fig. 2. (c) Integrated intensity of transverse
scans through scattering ridges, with the energy integrated between 2 meV
$\leq$ $\hbar\omega$ $\leq$ 6 meV. The red line is the Mn form factor squared.
Uncertainties throughout this paper are statistical and refer to one standard
deviation.

Figure 1 displays an intensity plot of $S(Q,\omega)$ at $T$ = 265 K in both
the ($H$ $K$ 0) and (0 $K$ $L$) scattering planes with the energy transfer
integrated between 3 meV $\leq$ $\hbar\omega$ $\leq$ 5 meV. This temperature
is in the paramagnetic phase of LCMO, at approximately 1.03 $T_{C}$. The most
prominent features of these data are circular rings of strong scattering
surrounding the Bragg positions. These rings can be attributed to spin
diffusive scatteringLynn et al. (1996) with an increase of the quasielastic
linewidth as $q$ moves away from the zone center.Chatterji et al. (2005);
Daoud-Aladine et al. (2006) For data with energy transfers from 5 meV to at
least 22 meV, the $q$ position corresponding to the maximum intensity of the
ring increases with the energy value of the constant-$E$ scan consistent with
the $\omega$ $\propto$ $q^{2.5}$ expectation of dynamical scaling theory;Lynn
(1984); Endoh and Hirota (1997); Halperin and Hohenberg (1969) in particular
the ring radii, as measured in scans along the ($\xi$ $\xi$ 0) direction, are
consistent with a quasielastic half width at half maximum (HWHM) that varies
with $q$ as $\Gamma(q)$ = $\Lambda q^{2.5}$ where $\Lambda$ = 18.9 $\pm$ 0.5
meV Å2.5. Closer to the zone boundary, the isotropy of the spin dynamics
breaks down and ridges of scattering are present which connect the rings along
($H$ 0 0) and equivalent directions. These ridges of scattering are strongest
in Brillouin zones at low $Q$ and the intensity falls off at higher $Q$ in a
manner roughly consistent with the Mn form factor squared, as displayed in
Fig. 1(c). These data reflect the integrated intensity of transverse scans
through ridges centered at positions equivalent to (0.5 0 0), (1 0.5 0), (1.5
0 0), (1 1 0.5), (1.5 1 0), and (2 0.5 0); all of the equivalent positions
within the range of the instrument were averaged. Given the strong coupling
between the magnetic and lattice degrees of freedom in CMR manganites it is
possible that these ridges of scattering also have a structural component;
with the current statistics we can only note that the $Q$ dependence suggests
that this scattering is primarily magnetic in origin.

Figure 2: (Color online) (a),(b) Differing intensities for scans in the (0
$\xi$ 0) and ($\xi$ $\xi$ 0) directions away from the (0 -1 0) reflection at
265 K. The blue data points and horizontal line correspond to the background
given by a scan in the ($\xi$ $\xi$ 0) direction at 100 K. (c) Scans through
the (0.5 1 0) position in both directions transverse to $\vec{q}$, measured at
265 K in the $\hbar\omega$ $\approx$ 3 meV data. (d) Transverse scan through
(0 0.4 0) in the elastic data, measured at 265 K. The lines in panels (c) and
(d) are Gaussian fits with a HWHM of 0.16 r.l.u. (0.26 Å-1). For panels
(a)-(c) the data have been integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$
directions transverse to the scan. For panel (d) the data have been integrated
over $\pm$ 0.13 Å-1 along $H$ and over $\pm$ 0.08 Å-1 along $K$.

These ridges of additional scattering are further explored in Fig. 2, where
scans in the (0 $\xi$ 0) and ($\xi$ $\xi$ 0) directions away from the (0 -1 0)
position are shown for energies centered at 3 meV [Fig. 2(a)] and 5 meV [Fig.
2(b)]. These scans are roughly isotropic through the ring, but when $q$
$\approx$ 0.55 Å-1 ($\approx$0.34 r.l.u. in the (0 $\xi$ 0) direction) the
data in the two directions diverge with the scattering in the ($\xi$ $\xi$ 0)
direction falling much faster. The width of these ridges of scattering in
directions transverse to $\vec{q}$ is about 0.26 Å-1 HWHM as shown in Fig.
2(c). Elastic or quasielastic scattering centered at $\vec{q}$ = (0.5 0 0) has
been observed in the CMR bilayer manganite La1.2Sr1.8Mn2O7Chatterji et al.
(2006) and the nearly half-doped manganite Pr0.55(Ca0.8Sr0.2)0.45MnO3Ye et al.
(2005) signifying short-range antiferromagnetic correlations. In
La0.7Ca0.3MnO3, the anomalous scattering near the (0.5 0 0) position instead
arises from a breakdown in dynamical scaling theory in which the energy width
of the quasielastic scattering ceases to be isotropic as $q$ approaches the
zone boundary, with smaller quasielastic widths for $\vec{q}$ values along the
($\xi$ 0 0) direction.

Figure 3: (Color online) (a) Energy dependence of the 265 K paramagnetic
scattering in constant-$Q$ scans. $Q$ values have been chosen so that
$\vec{q}$ away from (0 1 0) is along either the ($\xi$ 0 0) or ($\xi$ -$\xi$
0) directions, and with the magnitude of $q$ either 0.325 Å-1 or 0.812 Å-1.
The data have been folded across the $K$ = 0 plane and integrated over $\pm$
0.13 Å-1 in all $\vec{q}$ directions. (b) The energy dependence of the ridge
intensity, displayed as the difference in scattering for measurements at
$\vec{Q}$ = (0 0.6 0) and (0.283 0.717 0) (both with $q$ = 0.649 Å-1). The
gray line is a guide to the eye.

The intensity of this scattering anisotropy is energy-dependent, as shown in
the constant-$Q$ energy scans of Fig. 3(a). For smaller $q$ values where the
data are well described by simple spin diffusion, such as the $q$ = 0.325 Å-1
data shown in the figure, the energy scans do not depend on the orientation of
$\vec{q}$. In the higher-$q$ data, such as the $q$ = 0.812 Å-1 data shown in
the figure, the intensity of the ridge of extra scattering is demonstrated by
the difference in scattering for the $\vec{q}~{}||~{}(\xi~{}0~{}0)$ and
$\vec{q}$ $||$ ($\xi$ -$\xi$ 0) data. Figure 3(b) shows this difference in
scattering intensity for $q$ = 0.649 Å-1; this $\vec{q}$ position is shifted
slightly from the center of the ridge as the zone edge position will feature a
nuclear superlattice reflection in the elastic data. This anomalous scattering
near the zone edge is quasielastic in nature, having a maximum at the elastic
position and an energy HWHM of approximately 2.5 meV. This anisotropy ceases
to be measurable for $\hbar\omega$ $\gtrsim$ 15 meV; this is roughly the
energy transfer where rings of scattering surrounding adjacent Brillouin zone
centers begin to overlap. While this intensity is energy-dependent, the
transverse width of the ridges is not. A transverse scan through the ridge at
(0 0.4 0) is displayed in Fig. 2(d) for elastic scattering data (-1 meV $\leq$
$\hbar\omega$ $\leq$ 1 meV), showing a width (HWHM of 0.26 Å-1) equal to that
observed in the inelastic data.

Figure 4: (Color online) (a) Intensity plot of $S(Q,\,\omega)$ measured at 100
K, with the energy integrated between 24 meV $\leq$ $\hbar\omega$ $\leq$ 27
meV. $Q$ perpendicular to the scattering plane has been integrated over $\pm$
0.16 Å-1. The black and red dashed lines represent the scan directions in the
other panels. (b)-(c) Differing intensities for scans in the (0 $\xi$ 0) and
($\xi$ $\xi$ 0) directions away from the (1 0 0) reflection at 100 K. The data
have been integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$ directions
transverse to the scan.

These spin correlations above $T_{C}$ can be compared to the propagating spin
waves observed below $T_{C}$. Figure 4 displays data measured at 100 K. For
relatively low energies, such as $\hbar\omega$ $\approx$ 9 meV as shown in
Fig. 4(b), the spin waves are isotropic. At 100 K we do not observe any low-
energy anisotropy in $S(\vec{Q},\,\omega)$, in contrast to the anomalous
quasielastic ridges of scattering along the ($H$ 0 0) direction observed in
the paramagnetic phase. This is consistent with previous reportsLynn et al.
(1996) of the low-$q$ quasielastic scattering developing only at temperatures
approaching $T_{C}$. As expected, the widths of these spin waves in a
constant-$E$ scan (a HWHM of about 0.06 Å-1) are determined by the
instrumental resolution and are far narrower than the $Q$-space peaks arising
in the data above $T_{C}$. At higher energies, such as $\hbar\omega$ $\approx$
25.5 meV shown in Figs. 4(a) and 4(c), the peak in $Q$ space approaches the
zone boundary in the ($H$ 0 0) direction and thus the isotropy breaks down.
The constant-$E$ correlations in the ($H$ $K$ 0) plane develop into square
shapes which merge near the zone boundaries, consistent with the calculated
spin wave scattering.

Figure 5: (Color online) Scattering intensities with the energy integrated
between 24 meV $\leq$ $\hbar\omega$ $\leq$ 28 meV and $|L|$ = 0.5, measured at
265 K. (a) Intensity plot of the ($H$ $K$ 0.5) scattering plane. $Q$
perpendicular to the scattering plane has been integrated over $\pm$ 0.16 Å-1.
The black, red, and orange dashed lines represent the scan directions in the
other panels. (b) Differing intensities for scans in the ($\xi$ 0 0) and
($\xi$ $\xi$ 0) directions away from the (1 -1 0.5) reflection. (c) Scan in
the ($\xi$ 0 0) direction through the (1 -0.5 0.5) position. The Gaussian fit
has a HWHM of 0.16 r.l.u. (0.26 Å-1). The data in panels b and c have been
integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$ directions transverse to the
scan.

The presence of persistent spin correlations at 265 K can be further explored
through the spin dynamics in the ($H$ $K$ 0.5) scattering plane. For low
energy transfers, the spin correlations in this plane will present as
scattering centered at integer positions, arising from the ridges of
scattering displayed in Fig. 1. For higher energy transfers, such as 24 meV
$\leq$ $\hbar\omega$ $\leq$ 28 meV for the data shown in Fig. 5, the magnetic
scattering displays a more complex structure. To increase statistics the
measured data have been folded across the $L$ = 0 plane, so that data with $L$
= -0.5 and $L$ = 0.5 have been averaged together. The intensity plot of
correlations in the ($H$ $K$ 0.5) plane, shown as Fig. 5(a), is qualitatively
quite similar to the correlations in the ($H$ $K$ 0) plane at lower energies.
A similar anisotropy between scans along the ($\xi$ 0 0) and ($\xi$ $\xi$ 0)
directions is observed, as shown in Fig. 5(b); these ridges have a transverse
width, shown in Fig. 5(c), consistent with the low-energy data. It should be
noted that the ($H$ $K$ 0.5) scattering plane lies along a Brillouin zone
edge, so that all correlations in this plane are in the high-$q$ regime where
simple spin diffusion theory should not be applicable. In particular, the
correlations yield peaks in $q$ that are far narrower than would be expected
from a purely diffusive model. A breakdown in scaling theory in which the
width of peaks in constant-$E$ scans falls far below the theoretical value at
high $q$ was also observed in paramagnetic iron and nickel.Lynn (1984) These
correlations are also qualitatively similar to the ferromagnetic phase
$Q$-space spin wave correlations displayed in LCMO. The bilayer manganite
La1.2Sr1.8MnO7 was likewise reportedChatterji et al. (2005) to display
$Q$-space spin correlations in the paramagnetic phase that qualitatively
resembled those of ferromagnetic spin waves. Given the dispersion relation
displayed by propagating spin waves in LCMO below $T_{C}$, the $Q$-space
correlations in the ($H$ $K$ 0) plane at an energy of $\omega_{0}$ will have
the same structure as correlations in the ($H$ $K$ 0.5) plane at an energy of
$\omega_{0}$+$\Delta\omega$ where $\Delta\omega$ = $4S|J_{1}|$ (such that
$\Delta\omega$ $\approx$ 25 meV at 100 K). Finding the correlations of Fig. 5
in data where $\hbar\omega$ $\approx$ 26 meV suggests a significant
renormalization of $\Delta\omega$ from the 100 K data; this is reminiscent of
the previously reported renormalization of the spin wave stiffness,Lynn et al.
(1996, 1997) where $D(T_{C})$ $\approx$ $D(T=0)$/2.

It is clear that the spin correlations in LCMO above $T_{C}$ result in well-
defined peaks in constant-$E$ scans that qualitatively resemble the
correlations from the spin wave excitations below $T_{C}$; similar results
have been previously reported in a bilayer manganite.Chatterji et al. (2005)
Despite these well-defined peaks in constant-$E$ scans, no clear peaks at
finite energy are observed in constant-$Q$ scans. For $\vec{q}$ positions away
from the ($H$ 0 0) direction, the positions of these peaks are well described
by spin diffusive dynamical scaling theory. The data at higher-$q$ values
deviate from the expectations of dynamical scaling theory primarily through
peaks in the $Q$-space correlations that are far narrower than the simple spin
diffusive model would predict. An additional component of the paramagnetic
scattering is also observed as ridges of unexpectedly strong quasielastic
scattering at low energy transfers and $\vec{q}$ positions parallel to
$\vec{a}^{\ast}$ or a symmetry equivalent direction. Intrinsic inhomogeneity
with small scale regions of competing phases is a common signature in strongly
correlated electron materials, including stripe order in high-$T_{C}$
superconducting cupratesTranquada et al. (1995) and polar nanoregions in
relaxor ferroelectrics.Blinc et al. (1999) Phase separation of this sort is
well known in CMR manganites, with considerable evidence for lattice
polaronsLynn et al. (2007) and spin polarons;De Teresa et al. (1997) LCMO
samples with smaller doping levels also display evidence of ferromagnetic
droplets.Hennion et al. (1998) The physics of colossal magnetoresistance in
LCMO has been modeled as a percolationMayr et al. (2001) or Griffiths
phaseSalamon et al. (2002) effect arising from the separation of various
competing phases.

The effects of hole doping and applied field on the spin fluctuation spectrum
of LCMO near $T_{C}$ suggest that the small-$q$ spin diffusive portion of the
scattering arises from the short length-scale hopping of electrons on the
Mn3+/Mn4+ lattice;Lynn et al. (1996) this diffusive scattering coexists with
spin waves near $T_{C}$ and drives the ferromagnetic phase transition. It is
also known that the temperature dependence of the low-$q$ quasielastic
scattering is quite similar to the temperature dependences of the bulk
resistivity and the polaron correlations, suggesting a close connection
between paramagnetic scattering and colossal magnetoresistance. The new
high-$q$ quasielastic scattering in the paramagnetic phase described in this
work represents spin correlations with wavelengths approaching atomic length
scales, comparable in size to the small polarons generated by localized
electrons. We hope that further measurements on the short-range spin
correlations near $T_{C}$ will shed new light on the physics of colossal
magnetoresistance.

J.S.H. acknowledges support from the NRC/NIST Postdoctoral Associateship
Program. This research at Oak Ridge National Laboratory’s Spallation Neutron
Source was sponsored by the Scientific User Facilities Division, Office of
Basic Energy Sciences, U. S. Department of Energy.

$\ast$ email<EMAIL_ADDRESS>
${\dagger}$ email<EMAIL_ADDRESS>
## References

  * Wollan and Koehler (1955) E. O. Wollan and W. C. Koehler, Phys. Rev. 100, 545 (1955).
  * Schiffer et al. (1995) P. Schiffer, A. P. Ramirez, W. Bao, and S.-W. Cheong, Phys. Rev. Lett. 75, 3336 (1995).
  * Cheong and Chen (1998) S.-W. Cheong and C. H. Chen, in _Colossal Magnetoresistance, Charge Ordering and Related Properties of Manganese Oxides_ , edited by B. Raveau and C. N. R. Rao (World Scientific, Singapore, 1998).
  * Ramirez (1997) A. P. Ramirez, J. Phys.: Condens. Matter 9, 8171 (1997).
  * Millis et al. (1995) A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett. 74, 5144 (1995).
  * Röder et al. (1996) H. Röder, J. Zang, and A. R. Bishop, Phys. Rev. Lett. 76, 1356 (1996).
  * Varma (1996) C. M. Varma, Phys. Rev. B 54, 7328 (1996).
  * Dagotto (2003) E. Dagotto, _Nanoscale Phase Separation and Colossal Magnetoresistance_ (Springer, Berlin, 2003).
  * Dagotto (2005) E. Dagotto, Science 309, 257 (2005).
  * Lynn et al. (2007) J. W. Lynn, D. N. Argyriou, Y. Ren, Y. Chen, Y. M. Mukovskii, and D. A. Shulyatev, Phys. Rev. B 76, 014437 (2007).
  * Bridges et al. (2010) F. Bridges, L. Downward, J. J. Neumeier, and T. A. Tyson, Phys. Rev. B 81, 184401 (2010).
  * Goodenough (1955) J. B. Goodenough, Phys. Rev. 100, 564 (1955).
  * Lynn (2000) J. W. Lynn, J. Supercond. 13, 263 (2000).
  * Zhang et al. (2007) J. Zhang, F. Ye, H. Sha, P. Dai, J. A. Fernandez-Baca, and E. W. Plummer, J. Phys.: Condens. Matter 19, 315204 (2007).
  * Ye et al. (2007) F. Ye, P. Dai, J. A. Fernandez-Baca, D. T. Adroja, T. G. Perring, Y. Tomioka, and Y. Tokura, Phys. Rev. B 75, 144408 (2007).
  * Dai et al. (2000) P. Dai, H. Y. Hwang, J. Zhang, J. A. Fernandez-Baca, S.-W. Cheong, C. Kloc, Y. Tomioka, and Y. Tokura, Phys. Rev. B 61, 9553 (2000).
  * Lynn et al. (1996) J. W. Lynn, R. W. Erwin, J. A. Borchers, Q. Huang, A. Santoro, J.-L. Peng, and Z. Y. Li, Phys. Rev. Lett. 76, 4046 (1996).
  * Fernandez-Baca et al. (1998) J. A. Fernandez-Baca, P. Dai, H. Y. Hwang, C. Kloc, and S.-W. Cheong, Phys. Rev. Lett. 80, 4012 (1998).
  * Dai et al. (2001) P. Dai, J. A. Fernandez-Baca, E. W. Plummer, Y. Tomioka, and Y. Tokura, Phys. Rev. B 64, 224429 (2001).
  * Lynn et al. (1997) J. W. Lynn, R. W. Erwin, J. A. Borchers, A. Santoro, Q. Huang, J.-L. Peng, and R. L. Greene, J. Appl. Phys. 81, 5488 (1997).
  * Adams et al. (2000) C. P. Adams, J. W. Lynn, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, Phys. Rev. Lett. 85, 3954 (2000).
  * Adams et al. (2004) C. P. Adams, J. W. Lynn, V. N. Smolyaninova, A. Biswas, R. L. Greene, W. Ratcliff, S.-W. Cheong, Y. M. Mukovskii, and D. A. Shulyatev, Phys. Rev. B 70, 134414 (2004).
  * Shulyatev et al. (2002) D. Shulyatev, S. Karabashev, A. Arsenov, Y. Mukovskii, and S. Zverkov, J. Cryst. Growth 237-239, 810 (2002).
  * Chatterji et al. (2005) T. Chatterji, F. Demmel, G. Dhalenne, M.-A. Drouin, A. Revcolevschi, and R. Suryanarayanan, Phys. Rev. B 72, 014439 (2005).
  * Daoud-Aladine et al. (2006) A. Daoud-Aladine, B. Roessli, S. N. Gvasaliya, C. Perca, L. Pinsard-Gaudart, J. Rodríguez-Carvajal, and A. Revcolevschi, J. Phys.: Condens. Matter 18, 1509 (2006).
  * Lynn (1984) J. W. Lynn, Phys. Rev. Lett. 52, 775 (1984).
  * Endoh and Hirota (1997) Y. Endoh and K. Hirota, J. Phys. Soc. Jpn. 66, 2264 (1997).
  * Halperin and Hohenberg (1969) B. I. Halperin and P. C. Hohenberg, Phys. Rev. 177, 952 (1969).
  * Chatterji et al. (2006) T. Chatterji, M. M. Koza, F. Demmel, W. Schmidt, J.-U. Hoffmann, U. Aman, R. Schneider, G. Dhalenne, R. Suryanarayanan, and A. Revcolevschi, Phys. Rev. B 73, 104449 (2006).
  * Ye et al. (2005) F. Ye, J. A. Fernandez-Baca, P. Dai, J. W. Lynn, H. Kawano-Furukawa, H. Yoshizawa, Y. Tomioka, and Y. Tokura, Phys. Rev. B 72, 212404 (2005).
  * Tranquada et al. (1995) J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).
  * Blinc et al. (1999) R. Blinc, J. Dolinšek, A. Gregorovič, B. Zalar, C. Filipič, Z. Kutnjak, A. Levstik, and R. Pirc, Phys. Rev. Lett. 83, 424 (1999).
  * De Teresa et al. (1997) J. M. De Teresa, M. R. Ibarra, P. A. Algarabel, C. Ritter, C. Marquina, J. Blasco, J. García, A. del Moral, and Z. Arnold, Nature 386, 256 (1997).
  * Hennion et al. (1998) M. Hennion, F. Moussa, G. Biotteau, J. Rodríguez-Carvajal, L. Pinsard, and A. Revcolevschi, Phys. Rev. Lett. 81, 1957 (1998).
  * Mayr et al. (2001) M. Mayr, A. Moreo, J. A. Vergés, J. Arispe, A. Feiguin, and E. Dagotto, Phys. Rev. Lett. 86, 135 (2001).
  * Salamon et al. (2002) M. B. Salamon, P. Lin, and S. H. Chun, Phys. Rev. Lett. 88, 197203 (2002).