# ParaPose: Parameter and Domain Randomization Optimization for Pose
Estimation using Synthetic Data

Frederik Hagelskjær and Anders Glent Buch This project was funded in part by
Innovation Fund Denmark through the project MADE FAST, in part by the SDU
I4.0-Lab. Both authors are with SDU Robotics, Mærsk Mc-Kinney Møller
Institute, University of Southern Denmark, 5230 Odense M, Denmark
<EMAIL_ADDRESS>
###### Abstract

Pose estimation is the task of determining the 6D position of an object in a
scene. Pose estimation aid the abilities and flexibility of robotic set-ups.
However, the system must be configured towards the use case to perform
adequately. This configuration is time-consuming and limits the usability of
pose estimation and, thereby, robotic systems.

Deep learning is a method to overcome this configuration procedure by learning
parameters directly from the dataset. However, obtaining this training data
can also be very time-consuming. The use of synthetic training data avoids
this data collection problem, but a configuration of the training procedure is
necessary to overcome the domain gap problem. Additionally, the pose
estimation parameters also need to be configured. This configuration is
jokingly known as grad student descent as parameters are manually adjusted
until satisfactory results are obtained.

This paper presents a method for automatic configuration using only synthetic
data. This is accomplished by learning the domain randomization during network
training, and then using the domain randomization to optimize the pose
estimation parameters. The developed approach shows state-of-the-art
performance of 82.0 % recall on the challenging OCCLUSION dataset,
outperforming all previous methods with a large margin. These results prove
the validity of automatic set-up of pose estimation using purely synthetic
data.

##  I INTRODUCTION

The pose of a rigid object can be represented as a 6D transform consisting of
translation and rotation. This 6D pose allows a robotic set-up to manipulate
and analyze the object. By obtaining this pose through visual pose estimation,
the flexibility of the set-up is greatly increased. Without visual pose
estimation, the position of objects needs to be known beforehand by using
mechanical fixtures. Introducing new objects thus requires the assembly of new
fixtures.

(a) Synthetic training data

(b) Real test data

(c) No DR

(d) No DR

(e) With DR

(f) With DR

Figure 1:  Visualization of the domain randomization’s (DR) impact on the
network’s generalization ability. (a) and (b), show synthetic and real data,
respectively. (c) show a perfect feature prediction on synthetic data without
domain randomization. In (d), the network trained without domain randomization
shows bad results on the real test data. (e) show the more difficult feature
prediction with domain randomization, but increased performance on the test
scene in (f).

However, the complex configuration of the visual pose estimation system can
make the set-up very time-consuming. Different scenarios vary greatly, and it
is essential to fit the configuration to the scenario. Variations such as
camera type, lighting, background clutter, and levels of occlusion all
influence the system. While controlling these influences can simplify the set-
up [14], the environment cannot always be controlled. For example, attempting
to remove shadows in an image can quickly become a time-consuming task.
Properties such as object material and shape will also influence the
performance of the pose estimation algorithm. In academia, good results are
often obtained by the method known as Grad Student Descent [9], where hyper-
parameters are continuously tuned until adequate results are obtained.

As a result of the difficulties, the set-up time can often last more than a
week [12]. This set-up time significantly reduces the usability of robotics.
And as most of the time is spent on adjusting the software [12], it is
essential to simplify this part. One approach has been the automatic tuning of
pose estimation parameters using training data [13]. This method uses real
data to train the parameters. This approach has even shown increased
performance compared with parameters tuned manually. Deep learning algorithms
have used this approach very successfully, using real training data to tune
the network structure. On most benchmarking datasets, the top-performing
algorithms are based on deep neural networks [18, 1, 22, 36]. However,
obtaining real training is a task that needs to be performed manually.
Moreover, for deep learning models, this can easily amount to more than
hundreds of images that must be collected and correctly labeled. To overcome
this obstacle, the use of synthetic training data has been introduced [32, 4,
23]. By using synthetic training data, the manual set-up is greatly reduced,
while the algorithm retains high performance. In the BOP challenge [21]
methods using synthetic training data obtain results in the same range as
methods trained on real data. The method that the optimization in this paper
is based on also obtains results on the LINEMOD [18] and OCCLUSION [1]
datasets similar to methods trained on real data. However, the success of
methods trained on synthetic data is highly dependent on domain randomization
[10], which is configured manually. The pose estimation parameters are also
found heuristically to obtain the best results.

In this paper, we present a method for the automatic optimization of pose
estimation parameters. The optimization is performed using the same synthetic
data used for training the deep learning model. However, our results show that
domain randomization is necessary during parameter optimization to obtain good
results. The effect of the domain randomization is shown in Fig. 1. To this
end, we employ the domain randomization from the deep learning training. Using
domain randomization drastically improves the performance and allows our model
to outperform models trained with real data and heuristically found
parameters. On the challenging OCCLUSION [1] dataset, our method trained with
synthetic data, and optimized with synthetic data, obtains a recall of 82.0 %,
which is the new state-of-the-art performance.

To create a usable parameter optimization system, the parameters are split
into two groups. This allows the isolation of parameters that both influence
performance and run-time. For many applications, this run-time is very
important with respect to the feasibility of the system. Our optimization
allows the user to obtain the best performance to the desired maximum run-
time.

This paper presents the following main contributions.

  * •

A method for the automatic set-up of pose estimation systems using only
synthetic data.

  * •

A simple method to obtain the desired run-time of the system.

  * •

State-of-the-art results on the challenging OCCLUSION dataset.

The remaining paper is structured as follows: We first review related papers
in Sec. II. In Sec. III, our developed method is explained. In Sec. IV, the
parameter optimization is performed, and the performance is verified. Finally,
in Sec. V, a conclusion is given to this paper, and further work is discussed.

##  II RELATED WORK

Since the introduction of deep neural networks, domain randomization has
always been present [29]. Domain randomization is used to minimize the
distance between the training and test sets. The less the training set
represents the test set, the more critical domain randomization is.

The most common domain randomization strategies are adding Gaussian noise,
randomly cropping the image, rotating the image, and changing, hue, contrast
and brightness [29]. These methods are generally seen in many pose estimation
methods [23, 27, 16].

There also exist other strategies for the generalization of networks. One
commonly used strategy is pre-training, where the network is trained on a
large and diverse dataset to learn features, which are then re-used when
trained on the actual dataset. Another approach is the dropout of features
during training to avoid the network relying on a single feature.

####  II-1 Pose Estimation

Successful pose estimation methods are generally based on networks operating
on 2D images. This is a result of the success of 2D convolutional networks, as
many good networks have been developed with good pre-trained weights and
domain randomization strategies [15]. The following pose estimation methods
all use 2D networks for pose estimation. 3D information is then often used to
refine the poses. In our approach, we use the 3D data directly in the
learning. It gives the advantage that all modalities are included in the
network, but the disadvantage is that no networks pre-trained on large
networks are available.

An example is SSD-6D [23], a deep learning based pose estimation method using
only synthetic data. Renders of objects are cut and pasted onto images from
the COCO [26] dataset. The network is trained to classify the rotation and
presence of objects. Domain randomization is performed by changing the
brightness and contrast. The method is tested on the LINEMOD dataset, where it
obtains good performance but is outperformed by methods trained with real
data. In PVNet [27] 10000 images are rendered and cut and pasted onto images
from the SUN397 [37] dataset. Then random cropping, resizing, rotation, and
Gaussian noise is applied. This network also uses real training data along
with synthetic data. PVN3D [16] use the training data and domain randomization
from PVNet [27], but expand with 3D point-clouds. The network obtains very
good performance on the LINEMOD [18] dataset, which it is trained on, but on
the more challenging OCCLUSION dataset, it does not perform well. PoseCNN [36]
is the current state-of-the-art method on the OCCLUSION [1] dataset. Real data
is used along with synthetic renders with objects placed randomly in a scene.
The DPOD [39] method uses real training data and object renderings which are
cut and pasted onto images from the COCO [26] dataset. Brightness, saturation,
and Gaussian noise is used to increase the generalizability. Similar to our
method DenseFusion [34] process point-cloud data. The point-cloud processing
is combined with 2D processing, and the network is only trained on real data.
The network is tested on the LINEMOD [18] and obtains good results, but is
outperformed by methods using both using synthetic [10] or real data [16]. The
method CosyPose [24] show very good results with extensive domain
randomization. One million synthetic training images are generated, with
objects placed randomly. Gaussian blur, contrast, brightness, color, and
sharpness filters are all added to the images. The cut and paste strategy are
then used to increase the background variance. This data, along with real
training data, show state-of-the-art performance on two pose estimation
datasets. The method also showed the best recall for methods using only
synthetic data on the 2020 BOP challenge [21]. As mentioned, most of these
methods only use the cut and paste methods to generate the background. The
BlenderProc [4] dataset use photo-realistic renderings to include reflections
and shadows. However, methods using this dataset still use the cut and paste
methods to increase performance [24].

####  II-2 Learning Domain Randomization

In the reviewed pose estimation methods, the domain randomizations are
manually tuned. However, a number of methods exist to tune the domain
randomization automatically. One approach is to directly minimize the domain
gap between real and synthetic data by introducing unlabelled real data [8].
During training on the synthetic data, the learned features should not be able
to discriminate between real and synthetic data.

The domain randomization problem can also be seen as an optimization task
[40]. Here the COCO [26] dataset is used as a validation set where different
domain randomization strategies are tested. The best domain randomization
policy is then used for testing. The same strategy is seen in AutoAugment [3]
which tests different domain randomization strategies on a validation dataset.
Here state-of-the-art results are obtained on the many datasets.

Domain randomization can also be used to increase robustness against
adversarial images [38]. Similar to our method, the goal is to increase the
amount of domain randomization instead of finding the correct domain
randomization. They show that this improves performance compared with fixed
domain randomization.

####  II-3 Domain randomization in 3D

Successful deep learning of point-clouds was introduced with PointNet [28].
Here domain randomization consists of random Gaussian noise in the overall
position, Gaussian noise for each point, and a random rotation. We base our
domain randomization on this approach but add random Gaussian noise to the two
added channels of normal vector and RGB color. Additionally, as the image is
obtained with a 2.5D scanner, we add flattening of the point-cloud to simulate
wrongly obtained depth.

For 3D point-clouds a number of new domain randomization methods have also
been developed. One approach is cross-modal domain randomization [33], where
the object is cut and pasted into different scenes. More similar to the
approach seen in 2D, where more samples can be created from small amounts of
data. Another method to create more samples from existing data is point
interpolation [2]. New data is generated by interpolating between the points
from two different point-clouds. These two methods are useful when only small
amounts of data are present. Learning domain randomization for point-clouds
has also been performed. In PointAugment [25] a network is trained to generate
domain randomization. The domain randomization is both point-wise and shape-
wise. The augmentation is optimized during network training with the
requirement that the loss with augmentation should be slightly larger than
without augmentation. This approach shows improved performance compared with
classic augmentation. A method that successfully uses 3D domain randomization
is PointVoteNet [11], which is trained on real data. To create more samples
background data is cut and pasted.

####  II-4 Previous Method

Our pose estimation method is based on a previous method [10]. The method is
trained on the BlenderProc [4] dataset created for the BOP challenge [21].
This method use domain randomization based on the Kinect sensor. In this
paper, the domain randomization is optimized during the training without
knowledge of the scene.

An example of optimizing classic computer vision is seen with Bayesian
Optimization [13]. The parameters of an existing pose estimation method are
optimized using real data, and the improvement gives state-of-the-art
performance. We use a similar approach in our paper, but synthetic images with
domain randomization are used instead of real training data. Thus, the set-up
of our method is entirely based on synthetic data.

##  III METHOD

The pose estimation in this paper is based on an existing method [10]. In the
first step, a candidate detector is used, and from this output, the method
extracts point-clouds. These point-clouds are fed through a modified DGCNN
[35] network, and three predictions are obtained; classification, background
segmentation, and votes. The classification is the probability that the object
is in the center of the point-cloud. This is used to filter point-clouds, as
performing pose estimation for each point-cloud is very time-consuming. The
background segmentation and the votes are combined for the remaining point-
clouds to create keypoint matches to the object model. RANSAC [7] is then used
to find possible pose estimations, which are further refined by Coarse to Fine
Iterative Closest Point (C2F-ICP). Finally, the best pose estimation is
determined by a depth check using the projected model. Compared with the
previous method [10], the RANSAC, ICP, and depth distances are scaled by
object diagonal. The depth check is also aided by a contour check.

The deep neural networks are trained using only synthetic data from the
BlenderProc dataset [21, 4]. However, all parameters in the pose estimation
algorithm were originally found heuristically. In this paper, these parameters
are instead found by optimization, using the same synthetic data.

###  III-A Candidate Detector

In the original implementation, a MASK R-CNN [15] trained on the synthetic
BlenderProc [4] dataset was used as the candidate detector. However, the
network’s ability to classify point-clouds makes it agnostic to the candidate
detector. As a result, the method is also tested using detections from the
CosyPose method [24]. CosyPose was selected based on the good results using
synthetic data on the BOP datasets [24]. The detections have been created by
running the CosyPose detector trained on the BlenderProc dataset on the
complete LINEMOD [18] and OCCLUSION [1] datasets.

###  III-B Domain Randomization Optimization

Figure 2: Overview of the network training and domain randomization
optimization. The domain randomization model is updated after each epoch.

The method is based on a subset of domain randomizations, each with a noise
level. As in other methods [27], the level of noise in each sample is a random
Gaussian based on a selected max level. The task of the optimization is to
determine the max noise level for each subset of domain randomization, which
best improves the generalizability.

The six subsets of domain randomization are Gaussian XYZ noise, Gaussian
normal vector noise, Gaussian RGB noise, Gaussian RGB shift, rotation, and
flattening. The noise starts at a predetermined level of 1.0, 0.02, 0.02,
0.04, 5.0, and 0.02, respectively. Where point noise is in millimeter,
rotation is in degrees, and all other is in percentage. For each parameter,
the jump size is half the original value.

The first four epochs are run without domain randomization. After which, the
domain randomization is activated, with noise at predetermined levels. At
eight epochs, the optimization of the domain randomization is started by
recording the loss level and increasing the first noise level. After one
epoch, the loss is measured and compared with the recorded loss. If the loss
is decreased by more than 2.5 percent, then the noise isn’t limiting the
training, and the noise level is increased again. Otherwise, the next noise
type is increased. If the loss increase is more than five percent, the max
level is reduced to the previous level, and this noise type is no longer
increased. An overview of the training and domain randomization procedure is
shown in Fig. 2.

###  III-C Parameters

Several parameters are tuned in this pose estimation method to obtain good
performance. To explain the method, an overview is given of the optimized
parameters. For a further explanation of the method, the reader is referred to
the original paper [10]. We divide the parameters into continuous and discrete
parameters. The continuous parameters can have any real value within a known
range. An important aspect is that fine-tuning these values will primarily
increase the precision without influencing the run-time. The discrete
parameters are integer values that impact both the precision and run-time of
the algorithm. It is, therefore, essential to find the optimal trade-off for
the given task.

For the discrete parameters, the following parameters are optimized. The
number of point-clouds which are classified, $PC$. The remaining point-clouds
to perform pose estimation upon, $PE$. The number of RANSAC iterations, $RI$.
The number of RANSAC proposals to be sorted with the depth check, $DC$. And
the number of ICP iterations, $II$.

For the continuous parameters, the following are optimized. The cut-off radius
size to determine if less than 2048 points are accepted, $sr$. The threshold
for accepting votes at symmetric positions, $vt$. The distance in mm for the
RANSAC algorithm, $rd$. The C2F-ICP distance in mm, $id$. The scaling in the
C2F-ICP, $is$. The distance in mm to determine whether a point belongs to
foreground or object, $bd$. The distance for accepting object points, $ad$.

Additionally, a number of parameters are seen as essential to the structure of
the algorithm and are kept fixed. The C2F-ICP has three resolutions. The
network structure has 2048 input points. A minimum of 512 points is needed to
process a point-cloud, and a minimum of 100 key-point matches are necessary
for the RANSAC algorithm. The point-cloud is pre-processed to a 1mm voxel
grid. The object is pre-processed to a 5mm voxel grid for the C2F-ICP. The
object model is maximally at 100 keypoints on object model. The RANSAC
iterations are split into parts of 50. Normal vectors are computed with a
radius of 10 mm.

###  III-D Pose Estimation Parameter Optimization

As stated in Sec. III two different types of parameters are used in the pose
estimation method. The discrete parameters have a direct influence on the run-
time, while the continuous parameters do not. As a result of these
differences, two different parameter optimizations are performed.

First, the continuous parameters are found with fixed discrete parameter
values, and the optimized continuous parameters are then used for the discrete
optimization.

####  III-D1 Continuous Parameters

For the continuous parameters, the goal is to find the optimal parameter set
for performance. As many parameters are present, the parameters are expected
to influence each other. The risk of a local optimization finding a local
maximum is, therefore, present. To avoid this, a global optimization method is
used. The chosen optimization method is Bayesian Optimization with Upper
Confidence Bound as the acquisition function [30]. This method has been used
successfully for pose estimation parameter optimization [13]. It can be
represented by a single parameter, which is the trade-off between exploration
and exploitation. The method works without residuals, which is useful as very
small changes in parameter values will generally not change the recall. It is
a bounded optimization, and most values have logical bounds, and the remaining
can be bounded by reason, such as metric bounds.

The optimization consists of four iterations, with the first being random
exploration and then moving to an exploitation strategy. This is achieved by
decreasing the $\kappa$ value. 50 random iterations are performed, followed by
100 with $\kappa=0.5$, then 50 with $\kappa=0.1$, and finally 50 with
$\kappa=0.01$, resulting in 250 iterations in all.

####  III-D2 Discrete Parameters

The discrete parameter values both influence the recall and the run-time. As a
result, one cannot simply determine the best parameter set. Instead, the goal
of this optimization is to find a list of parameter sets. This list will give
the parameter sets that give the highest increase in recall when increasing
the run-time. A user would then be able to select the best trade-off for the
given situation.

As the parameter values are discrete, the optimization is performed with a
grid search. All parameter sets are sorted according to run-time, and starting
with the lowest run-time, parameter sets are added if they increase the
recall.

####  III-D3 Varying Number of Objects

The algorithm’s run-time depends not only on the parameter values but also on
the number of searched objects. While a new optimization could be performed if
any objects are removed or added, it is desirable to perform the optimization
once and then adjust the parameters to fit run-time given the number of
objects. The relationship between discrete parameter values, number of
objects, and run-time are shown in Eq. 1. This equation can be solved using
least-squares on the grid search results. Here $t_{image}$ is the run-time of
the system, $t_{pre}$ is the pre-processing and initial detection of interest
points, $t_{net}$ is the is run-time of the network, $t_{ran}$ is the run-time
of a RANSAC iteration, $t_{icp}$ is the run-time of an ICP iteration, and
$t_{depth}$ is the run-time of the depth check.

$\displaystyle t_{image}=t_{pre}+obj*(t_{net}*PC+PE*$ (1)
$\displaystyle(t_{ran}*RI+DC*(t_{icp}*II+t_{depth})))$

###  III-E Evaluation Metrics

The success criteria of a pose estimation must first be defined before the
performance can be optimized.

A simple score is the distance between the ground truth pose and the pose
estimate. This is performed in the Tejani [31] and Doumanoglou [5] datasets
poses with less than 50mm error are declared correct. This metric also adds a
fifteen-degree maximum error between the Z-axes to include angular errors
while accommodating symmetric objects. Two central problems with this score
are that errors further away from the object center are penalized less
severely, and only angular errors in the Z-axis are included. For the LINEMOD
[18] and OCCLUSION [1] datasets, the average distance (ADD) metric is used.
First, the object model is converted to a point-cloud which is projected into
the scene both by the ground truth and the found pose. The ADD metric is then
computed as the average distance between each point in the two point-clouds. A
pose estimate is successful if the average distance is smaller than ten
percent of the object model’s diagonal. The ADD/I score can be used to
accommodate symmetric objects where the distance is simply to the nearest
point in the comparison cloud. As the ground truth labels in the OCCLUSION
dataset are not very precise, this metric fits well with the dataset. And as
the test datasets use this score for evaluation, it would seem obvious to use
this for the optimization. However, as the parameter optimization is performed
with synthetic data, the ground truth is exact, and a more discriminating
metric can be used. Therefore, the metric established from the BOP test is
used for the optimization. The metric consists of three different scores,
Visible Surface Discrepancy (VSD) [20, 19], Maximum Symmetry-Aware Surface
Distance (MSSD) [6], and Maximum Symmetry-Aware Projection Distance (MSPD)
[21]. To understand the metric a very general description is given for each
score. The object is first projected into the scene according to the ground
truth pose and the pose estimate. In VSD, the distance in depth, for the
visible part of the model is calculated. In MSSD, the maximum distance between
each point in the model is calculated. The last score, MSPD, is the distance
in pixels of the projected object and is therefore good for methods without
depth data. Multiple thresholds are used instead of a single cut-off value,
and the multiple scores are combined into an average.

##  IV EXPERIMENTS

To show the performance of the developed method, tests are performed on the
LINEMOD dataset [18] and the challenging OCCLUSION [1] dataset. The parameters
are first optimized, and the optimized method is then tested on the real data
and compared with the previous approach and current state-of-the-art methods.
The optimization is performed with the synthetic BlenderProc dataset [21, 4].
For the network training, 10000 synthetic images are used, and for the
parameter optimization, nine previously unseen synthetic images are used. All
experiments are performed on a PC environment (Intel i9-9820X 3.30GHz CPU and
NVIDIA GeForce RTX 2080 GPU).

###  IV-A Network and Domain Randomization Optimization

For each object, a network and domain randomization is trained. The training
runs for sixty epochs with parameters according to the previous method [10].
The resulting noise levels for three objects are shown in Fig. 3. The objects
are Ape, Can, and Eggbox, as they respectively represent small, large, and
symmetric objects. It is seen that the network tolerates a much lower noise
increase for XYZ and rotation. The noise levels appear similar for all
objects, although the symmetric object Eggbox has a very low XYZ level.

###  IV-B Continuous Optimization

The continuous optimization is performed using Bayesian Optimization. The
discrete parameters are fixed to values of, $PC=32$, $PE=6$, $RI=500$,
$II=10$, and $DC=2$. By setting $PE=6$ and $DC=2$, twelve pose estimations are
checked for each object. This ensures that all the continuous parameters are
used for the pose estimation without making the run-time unfeasible. The
optimization is performed both with and without domain randomization.
Additionally, an optimization is made using the ADD/I metric instead of the
BOP. The algorithm’s recall resulting from the parameters span from 40 % to 93
%. The correct parameters thus have a strong influence on the performance. The
parameters with and without domain randomization, and the heuristically found
parameters, are shown in Tab. I. Compared with the heuristic parameters, a
notable difference is seen with the much lower voting threshold, $vt$, giving
more matches to RANSAC. Compared with the optimization without domain
randomization, the C2F-ICP and depth check parameters are the most significant
difference. Here a much more refined search is performed compared with using
domain randomization. The same tendency is seen with the optimization using
the ADD/I metric.

Figure 3: Number of noise level increases for each noise type shown for three different types of objects.  TABLE I: The found values from the continuous parameter search compared with the heuristic parameters. Param | $vt$ | $rd$ | $id$ | $is$ | $bd$ | $ad$ | $sr$
---|---|---|---|---|---|---|---
Heur. | 0.95 | 10.0 | 2.5 | 2.0 | 10 | 5 | 72
No DR Opt. | 0.27 | 17.03 | 1.24 | 2.25 | 21 | 1 | 108
Opt. ADD | 0.275 | 12.85 | 0.77 | 3.49 | 59 | 5 | 66
Opt. | 0.174 | 19.88 | 4.85 | 1.24 | 86 | 12 | 108

###  IV-C Discrete Optimization

The discrete optimization is performed with the parameters found in the
continuous optimization. A grid search is then performed using varying
discrete parameters. The tested values are shown in Tab. II. As unfeasible
parameter settings cannot be tested, e.g., $PE=10$ when $PC=8$, the final test
iterations are 576. As the optimization is performed with 15 objects, the run-
time will not match the OCCLUSION dataset, which has only eight objects. To
compute the expected run-time, the 576 iterations from the discrete
optimization are used to solve Eq. 1 using least-square optimization. The
resulting time contribution by increasing each discrete parameter is shown in
Tab. III.

TABLE II: Values tested in the discrete grid search. $PC$ | $PE$ | $RI$ | $DC$ | $II$
---|---|---|---|---
8,16,32 | 2,4,6,8,10 | 500,1500,2500 | 1,2,5,10 | 10,30,50
TABLE III: Time consumption of each part of the model. Part | $t_{pre}$ | $t_{net}$ | $t_{ran}$ | $t_{depth}$ | $t_{icp}$
---|---|---|---|---|---
Time (s) | 8.57e-01 | 7.99e-03 | 2.70e-04 | 9.12e-03 | 1.67e-04
Figure 4:  The output of the discrete optimization shown as blue crosses. The
red line shows the best-performing parameter sets in relation to run-time.
Parameter values are visualized for some of the parameters which have the best
trade-off between run-time and performance.

The results are sorted according to run-time and performance, and the results
are shown in Fig. 4. Of the 576 iterations, 14 parameter sets give the best
trade-off between run-time and performance. The least influential parameter is
found to be the number of C2F-ICP iterations, which is always at 10. While the
depth check improves the performance, it also significantly increases the run-
time. It is, therefore, only increased in value at the maximum run-time.

TABLE IV: The resulting run-time and recall for the heuristic approach and with optimized parameters, with different run-times according to Fig. 4. Results are shown with the original MASK R-CNN detector and the CosyPose detector.  Method | Run-time (s) |  Recall ($\%$)
---|---|---
Previous method | 3.5 | 77.2
Heuristic $<$4 sec (prev. net) | 3.4 | 77.58
Heuristic $<$4 sec | 3.4 | 77.63
No DR Opt. $<$4 sec | 3.7 | 77.40
ADD/I Opt. $<$4 sec | 3.6 | 77.49
Opt. $<$4 sec, MASK R-CNN | 3.6 | 78.91
Opt. Max, MASK R-CNN | 26.3 | 80.11
Opt. $<$4 sec, CosyPose | 3.6 | 80.45
Opt. Max, CosyPose | 26.2 | 82.03
TABLE V:  Results for the LINEMOD dataset [18] in % recall with the ADD/I score. The competing methods are DPOD [39], SSD-6D [23] (obtained from [34]), DenseFusion [34], PointVoteNet [11], PVN3D [16] and the previous method [10]. Rotation invariant objects are marked with an *.  | Training
---
Data
Real | Synthetic
| DenseFusion | DPOD | PointVoteNet | PVN3D | DPOD | SSD6D | Prev. | ParaPose
|  |  |  |  |  |  |  | $<4s$ | MAX
Ape | 92 | 87.7 | 80.7 | 97.3 | 55.2 | 65 | 97.7 | 97.9 | 97.9
Bench v. | 93 | 98.5 | 100 | 99.7 | 72.7 | 80 | 99.8 | 100 | 100
Camera | 94 | 96.1 | 100 | 99.6 | 34.8 | 78 | 98.3 | 97.9 | 97.9
Can | 93 | 99.7 | 99.7 | 99.5 | 83.6 | 86 | 98.8 | 99.9 | 99.9
Cat | 97 | 94.7 | 99.8 | 99.8 | 65.1 | 70 | 99.9 | 99.9 | 99.8
Driller | 87 | 98.8 | 99.9 | 99.3 | 73.3 | 73 | 99.2 | 100 | 100
Duck | 92 | 86.3 | 97.9 | 98.2 | 50.0 | 66 | 97.8 | 98.3 | 98.3
Eggbox* | 100 | 99.9 | 99.9 | 99.8 | 89.1 | 100 | 97.7 | 98.5 | 98.5
Glue* | 100 | 96.8 | 84.4 | 100 | 84.4 | 100 | 98.9 | 99.9 | 100
Hole p. | 92 | 86.9 | 92.8 | 99.9 | 35.4 | 49 | 94.1 | 98.8 | 99.5
Iron | 97 | 100 | 100 | 99.7 | 98.8 | 78 | 100 | 100 | 100
Lamp | 95 | 96.8 | 100 | 99.8 | 74.3 | 73 | 92.8 | 100 | 99.9
Phone | 93 | 94.7 | 96.2 | 99.5 | 47.0 | 79 | 99.1 | 99.3 | 99.4
Average | 94.3 | 95.15 | 96.3 | 99.4 | 66.4 | 79 | 98.0 | 99.3 | 99.3

###  IV-D Performance on real data

Several experiments are performed to test the effectiveness of the synthetic
parameter optimization method. All tests are performed on the OCCLUSION [1]
dataset. The tests are performed with the parameters found in the parameter
optimization. For the discrete parameters, two-parameter sets are selected,
one for a run-time under four seconds and one with maximum run-time. The
parameters are ($PC=16,PE=4,RI=1500,II=10,DC=1$) and
($PC=32,PE=8,RI=2500,II=10,DC=10$), respectively.

We also test the previously trained network with our modified method, without
the parameter optimization but with the same discrete parameters.
Additionally, we test the new network with heuristic parameters and with
parameter optimization performed without domain randomization. For the
optimized parameters, tests are performed both with the MASK R-CNN detector
and the CosyPose detector. The resulting recall and run-times of the tests are
shown in Tab. IV. The results show that the network using optimized domain
randomization has the same performance as the manually tuned model but does
not significantly increase the performance. The optimization without domain
randomization shows poor generalization and performs worse than heuristically
tuned parameters. However, the significance of the parameter optimization with
domain randomization is shown. The same method increases from a recall of
77.63 % to 78.91 % by tuning the parameters. The optimization using the ADD/I
metric shows poor results, verifying the importance of the BOP metric. The
maximum run-time parameters are also shown to increase the performance, going
to a recall of 80.11 %. Additionally, using the CosyPose detector gives
superior results, increasing the recall to 82.0 %.

###  IV-E Comparison with state-of-the-art methods

TABLE VI:  Results on the OCCLUSION dataset [1] in % recall with the ADD/I score. The compared methods are PointVoteNet [11], PVN3D [16], PoseCNN [36], and the previous method [10]. The score for [16] is from [17]. Rotation invariant objects are marked with an *. | Training
---
Data
Real | Synthetic
| [11] | [16] | [36] | Prev. | ParaPose
|  |  |  |  |  $<$ 4 s | MAX
Ape | 70.0 | 33.9 | 76.2 | 66.1 | 66.6 | 68.5
Can | 95.5 | 88.6 | 87.4 | 91.5 | 94.3 | 95.7
Cat | 60.8 | 39.1 | 52.2 | 60.7 | 66.3 | 68.8
Driller | 87.9 | 78.4 | 90.3 | 92.8 | 92.8 | 94.6
Duck | 70.7 | 41.9 | 77.7 | 71.2 | 73.7 | 74.8
Eggbox* | 58.7 | 80.9 | 72.2 | 69.7 | 71.3 | 73.4
Glue* | 66.9 | 68.1 | 76.7 | 71.5 | 81.1 | 81.8
Hole p. | 90.6 | 74.7 | 91.4 | 91.5 | 96.4 | 97.3
Average | 75.1 | 63.2 | 78.0 | 77.2 | 80.5 | 82.0

The method is compared with current state-of-the-art methods to show the
effectiveness of the developed parameter optimization. Only methods using
RGB-D data are included as these have the highest performance. We showcase the
results using the CosyPose [24] detector with the optimized parameters for
run-time under four seconds and maximum run-time. Results are shown for the
LINEMOD and the OCCLUSION datasets. The performance on LINEMOD is shown in
Tab. V. The results show that our method outperforms all methods trained on
synthetic data and most methods using real training data, while only PVN3D
[16] obtain better results. The results for the OCCLUSION dataset are shown in
Tab. VI. Apart from the previous method [10], all methods have been trained on
real data. Our methods outperforms all previous methods and obtains a recall
of 82.0 %. This is the new state-of-the-art result on this dataset, an
improvement of four percentage points, and is obtained by using only synthetic
data. Additionally, it should be noted that the method PVN3D [16] which
obtained the highest score on the LINEMOD [18] dataset, only obtains a 63.2 %
recall on this more challenging dataset.

##  V CONCLUSION

This paper presents a method for the set-up of pose estimation using only
synthetic data. The network is trained, and parameters are optimized using
synthetic data. The method also provides the best trade-off between
performance to a given run-time. While the run-time is estimated using
simulation, it generally matches the real world and allows for a more
straightforward set-up procedure.

By using the optimized parameters, the pose estimation system outperforms the
original method using manually tuned parameters. The developed method is
compared with current state-of-the-art methods, which also use real training
data. On the challenging OCCLUSION dataset, our method obtains a recall of
82.0 %, which is the new state-of-the-art.

In future work, this method could be expanded to process entire scene point-
clouds. The network can then be pre-trained on large datasets to create strong
weights which increase generalizability. The network can also be trained for
multiple objects, and the noise model can be trained for the entire scene.

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T\rrbracket$, the pro-$p$ ring of all formal power series over the field of
$p$ elements, which is a local ring with unique maximal ideal
$\mathfrak{m}=T\,\mathbb{F}_{p}\llbracket T\rrbracket$ generated by the
indeterminate $T$. Then take $G=\mathrm{SL}_{n}^{1}(R)$, the first principal
congruence subgroup of the special linear group of all $n\times n$ matrices of
determinant $1$ over $R$; that is,

$G=\\{\,g\in\mathrm{SL}_{n}(R)\mid g\equiv I\pmod{\mathfrak{m}}\,\\},$

where $I$ denotes the $n\times n$ identity matrix. Using the technology of
[19], it is straightforward to observe that $G$ is a hereditarily just
infinite pro-$p$ group. First $G$ is $R$-perfect and so the terms of its lower
central series are the congruence subgroups

$\gamma_{k}(G)=G_{k}=\\{\,g\in\mathrm{SL}_{n}(R)\mid g\equiv
I\pmod{\mathfrak{m}^{k}}\,\\},$

for each $k\geqslant 1$. Adapting slightly the notation used in [19], we see
that the (completed) graded Lie ring associated to the lower central series of
$G$ satisfies

$L(G)=L_{G}(G)=\overline{\bigoplus_{i=1}^{\infty}G_{i}/G_{i+1}}\cong\prod_{i=1}^{\infty}T^{i}\mathfrak{sl}_{n}(\mathbb{F}_{p})\cong\mathfrak{sl}_{n}(\mathfrak{m}),$

the latter being the Lie algebra over $\mathbb{F}_{p}$ of $n\times n$ matrices
with entries in $\mathfrak{m}$ and trace $0$. To every closed subgroup $H$ of
$G$ we associate a closed Lie subalgebra of $L(G)$ that we denote by
$L_{G}(H)$ and whose properties are described in [19, Lemma 2.13]. Using the
isomorphism above we view $L_{G}(H)$ as a Lie subalgebra of
$\mathfrak{sl}_{n}(\mathfrak{m})$. In particular, $L_{G}(G_{k})$ corresponds
to the Lie subalgebra
$\prod_{i=k}^{\infty}T^{i}\mathfrak{sl}_{n}(\mathbb{F}_{p})\cong\mathfrak{sl}_{n}(\mathfrak{m}^{k})$.
If $W$ is a non-zero $\mathbb{F}_{p}$-subspace of $\mathfrak{sl}_{n}(R)$
satisfying $[W,\mathfrak{sl}_{n}(\mathfrak{m}^{k})]_{\text{Lie}}\subseteq W$
for some $k\geqslant 1$, then a direct computation shows there exists $r$ such
that $\mathfrak{sl}_{n}(\mathfrak{m}^{r})\subseteq W$. (It is this computation
that uses the fact that $p\nmid n$.)

Now let $H$ and $N$ be closed subgroups of $G$ such that $\mathbf{1}\neq
N\trianglelefteqslant H\trianglelefteqslant G$. Then $L_{G}(H)$ is an ideal of
the Lie algebra $\mathfrak{sl}_{n}(\mathfrak{m})$ and hence there exists $r$
such that $\mathfrak{sl}_{n}(\mathfrak{m}^{r})\subseteq L_{G}(H)$.
Consequently, $G_{r}\leqslant H$, so that $[N,G_{r}]\leqslant N$ and one
deduces $[L_{G}(N),\mathfrak{sl}_{n}(\mathfrak{m}^{r})]_{\text{Lie}}\subseteq
L_{G}(N)$. It follows that $\mathfrak{sl}_{n}(\mathfrak{m}^{s})\subseteq
L_{G}(N)$ for some $s$ and hence $G_{s}\leqslant N$ and so
$\mathopen{|}G:N\mathclose{|}<\infty$. This demonstrates that $G$ is
hereditarily just infinite.

Next we exploit properties of the Nottingham group $\mathcal{N}$ over
$\mathbb{F}_{p}$ to produce groups of automorphisms of the above group $G$.
The group $\mathcal{N}$ is the Sylow pro-$p$ subgroup of the profinite group
$\operatorname{Aut}_{\mathrm{c}}(R)=\operatorname{Aut}(R)$; it coincides with
the group $\operatorname{Aut}^{1}(R)$ of all automorphisms of the ring $R$
that act trivially modulo $\mathfrak{m}^{2}$. Any element $\alpha$ of
$\mathcal{N}$ is then uniquely determined by its effect upon the indeterminate
$T$ and, conversely, for any $f\in R$ with $f\equiv T\pmod{\mathfrak{m}^{2}}$
there is a unique element of $\mathcal{N}$ mapping $T$ to $f$. (Thus
$\mathcal{N}$ could alternatively be defined as a group of power series
$T+\mathfrak{m}^{2}$ with the binary operation given by substitution of power
series. For our construction, however, the behaviour as automorphisms is more
relevant.) We refer to [5] for background material concerning the Nottingham
group, which plays a role also in number theory and dynamics.

The action of the Nottingham group $\mathcal{N}$ on $R$ induces a faithful
action upon the group $G=\mathrm{SL}_{n}^{1}(R)$ and hence we construct a
subgroup $\dot{\mathcal{N}}\leqslant\operatorname{Aut}_{\mathrm{c}}G$
isomorphic to $\mathcal{N}$. Suppose $\alpha\in\mathcal{N}$ is an element that
induces an inner automorphism $\dot{\alpha}$ of the group $G$, and put
$f=T\alpha\in T+\mathfrak{m}^{2}$. Then there exists a matrix $h\in G$ such
that $hx^{\dot{\alpha}}=xh$ for all $x\in G$. In particular, upon taking
$x=I+Te_{ij}$ for $1\leqslant i,j\leqslant n$ with $i\neq j$, we conclude that
$h$ must be a diagonal matrix such that every pair of distinct diagonal
entries $a$ and $b$ are linked by the relation $Ta=fb$ in $R$. It follows that
$f^{2}=T^{2}$ and hence, since $f\equiv T\pmod{\mathfrak{m}^{2}}$, that $f=T$
and $\dot{\alpha}=\mathrm{id}_{G}$. In conclusion, the copy of the Nottingham
group in $\operatorname{Aut}_{\mathrm{c}}G$ satisfies
$\dot{\mathcal{N}}\cap\operatorname{Inn}G=\mathbf{1}$.

As the final step in our construction, we use the result of Camina [4] that
every countably-based pro-$p$ group can be embedded as a closed subgroup in
$\mathcal{N}$. Hence if $A$ is any finitely generated pro-$p$ group that is
virtually nilpotent (say, of class $c$), then it may be embedded in
$\operatorname{Aut}_{\mathrm{c}}G$ in such a way that
$A\cap\operatorname{Inn}G=\mathbf{1}$. Hence we have satisfied the conditions
of Theorem 6.2 and the semidirect product $G\rtimes A$ is an example of a
hereditarily JNNcF pro-$p$ group.

#### Acknowledgements

Some of this research was conducted while the second author was visiting
Heinrich Heine University Düsseldorf. He thanks the university for its
hospitality and we gratefully acknowledge partial support by the Humboldt
Foundation. We also thank the referees for their careful reading of the paper
and insightful suggestions, particularly that of using subprimitivity in
Section 6.

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Benjamin Klopsch<EMAIL_ADDRESS>
Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
Martyn Quick<EMAIL_ADDRESS>
School of Mathematics & Statistics, University of St Andrews, St Andrews, UK


Search for rare decays and lepton-flavor-violating decays
of the Higgs boson at the ATLAS experiment

Pawel Bruckman de Renstrom
on behalf of the ATLAS Collaboration

Institute of Nuclear Physics PAN, Cracow, Poland

> The Standard Model predicts several rare Higgs boson decay channels, among
> which are the decays to a Z boson and a photon, to a low-mass lepton pair
> and a photon, and to a meson and a photon. The observation of some of these
> decays could open the possibility of studying the CP and coupling properties
> of the Higgs boson in a complementary way to other analyses. In addition,
> lepton-flavor-violating decays of the observed Higgs boson are searched for,
> where an observation would be a clear sign of physics effects beyond the
> Standard Model. These proceedings present selected recent results for such
> decays based on proton–proton collision data at $13$ TeV collected by the
> ATLAS experiment in Run 2 of the LHC.

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

Copyright [2023] CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0
license.

##  1 Introduction

Since the discovery of the Higgs boson by the CMS and ATLAS Collaborations in
2012, its properties have been measured to increasing precision. So far, an
excellent agreement with the predictions for a Standard Model (SM) Higgs boson
is observed. However, the SM, while highly successful, is not considered to be
a complete theory as it is not capable of explaining some of the phenomena
seen in nature. Studying Higgs decays which are highly suppressed or forbidden
in the SM is particularly attractive to probe for the smoking gun of phenomena
from beyond the SM. In the following, three categories of search analyses are
reported. The first concerns with Higgs decays occurring via loop diagrams or
suppressed by the small Yukawa coupling constant to the first or the second
generation fermions. The second one are searches for lepton flavour violating
(LFV) decays of the Higgs boson. The third category are Higgs decays to the
first or the second generation leptons. The proceedings summarizes selected
recent search results, obtained using proton–proton ($pp$) collision data at
$13$ TeV collected by the ATLAS experiment [1] in Run 2 of the LHC. All
presented results concern rare processes and are statistically limited. Hence,
no discussion of systematic uncertainties is included in the following.

##  2 Search for rare loop-induced or Yukawa-suppressed Higgs decays

Higgs decays to quarkonia or light mesons with an emission of a prompt photon
occur via loop diagrams or direct diagrams suppressed by the small Yukawa
coupling constant to the first or the second generation fermions. The
branching fraction expected in the SM may be additionally suppressed due to
strong destructive interference of the loop and the direct decay diagrams.
Modifications to either Yukawa couplings or additional contributions to the
loop diagrams could substantially enhance the branching.
$H\rightarrow Z\gamma$ with leptonic $Z$ decays
The SM Higgs boson can decay into $Z\gamma$ through loop diagrams and the
branching ratio is predicted to be $B(H\rightarrow Z\gamma)=(1.54\pm
0.09)\times 10^{-3}$ at $m_{H}=125.09{\mathrm{\ Ge\kern-1.00006ptV}}$ [2]. It
can differ from the SM value for several scenarios beyond the SM, for example,
if the Higgs boson was a neutral scalar of different origin, or a composite
state. Different branching ratios are also expected for models with additional
colourless charged scalars, leptons or vector bosons that couple to the Higgs
boson, due to their contributions via loop corrections. A search for $Z\gamma$
decays of the SM Higgs boson in 139 fb-1 of $pp$ collisions at
$\sqrt{s}=13{\mathrm{\ Te\kern-1.00006ptV}}$ is performed in the leptonic $Z$
decay channel, where lepton accounts for either an electron or a muon [3]. The
invariant mass $m_{Z\gamma}$ is reconstructed using kinematical fit with the
$Z$ mass constraint on the emerging lepton pair. The resulting mass spectrum
is symultaeously fitted in six mutually exclusive signal regions. The weighted
sum of all the region together with the fit results is shown in Fig. 1(a). The
observed data are consistent with the expected background with the $p$-value
of 1.3%, while the expected $p$-value in the presence of the SM Higgs boson is
12.3%, corresponding to significances of 2.2 and 1.2 $\sigma$, respectively.
The observed 95% confidence level (CL) upper limit on the
$\sigma(pp\rightarrow H)\times B(H\rightarrow Z\gamma)$ is 3.6 times the SM
prediction for the Higgs boson mass of 125.09 GeV. The expected limit on
$\sigma(pp\rightarrow H)\times B(H\rightarrow Z\gamma)$ assuming either no
Higgs boson decay into $Z\gamma$ or the presence of the SM Higgs boson decay
is 1.7 and 2.6 times the SM prediction, respectively. The best-fit value for
the signal yield normalised to the SM prediction is $2.0^{+1.0}_{-0.9}$ where
the statistical component of the uncertainty is dominant.

(a) $H\rightarrow Z\gamma$ selection [3]

(b) $H\rightarrow\ell\ell\gamma$ selection [4]

Figure 1:  Invariant mass distribution of events satisfying signal selection
criteria in data and fit results while fitting all signal categories
simultaneously. Bottom panels show the residuals of the data with respect to
the non-resonant background component of the signal plus background fit.

$H\rightarrow\gamma^{*}\gamma$ with $\gamma^{*}$ decaying into a low mass
lepton pair
The analysis targets quasi the same final state as the one discussed above,
but the requirement of low invariant mass of the lepton pair
($m_{\ell\ell}<30{\mathrm{\ Ge\kern-1.00006ptV}}$) makes this signal region
completely dominated by the decay through a $\gamma^{*}$ [4]. With the
requirement of low $\ell\ell$ mass and high $p_{\rm T}$ electron clusters can
overlap, hence the analysis introduces an additional signal category of merged
electrons. Consequently, a dedicated reconstruction of merged electron pairs,
based on machine learning methods is put in place. The weighted sum of the
resulting 9 mutually exclusive signal regions together with the fit result is
shown in Fig. 1(b). For a Higgs boson with a mass of 125.09 GeV and
$m_{\ell\ell}<30{\mathrm{\ Ge\kern-1.00006ptV}}$, evidence for the
$H\rightarrow\ell\ell\gamma$ process is found with the significance of
$3.2\sigma$ over the background-only hypothesis, compared to an expected
significance of $2.1\sigma$. The best-fit value of the signal-strength
parameter, defined as the ratio of the observed signal yield to the signal
yield expected in the SM, is $\mu=1.5\pm 0.5$. The Higgs boson production
cross-section times the $H\rightarrow\ell\ell\gamma$ branching ratio for
$m_{\ell\ell}<30{\mathrm{\ Ge\kern-1.00006ptV}}$ is determined to be
$8.7^{+2.8}_{-2.7}\ {\mathrm{fb}}$. This result constitutes the first evidence
for the decay of the Higgs boson into a pair of leptons and a photon, an
important step towards probing Higgs boson couplings in this rare decay
channel.

Higgs decays to exclusivly reconstructed vector quarkonia or light mesons and
a photon
Due to highly destrucive interference between the indirect and direct Feynman
diagrams, Higgs boson decays in the charmonium sector, $H\rightarrow
J/\psi\gamma$ and $H\rightarrow\psi(2S)\gamma$, offer an opportunity to access
both the magnitude and the sign of the charm-quark Yukawa coupling; the
corresponding decays in the bottomonium sector,
$H\rightarrow\Upsilon(1S,2S,3S)\gamma$, can provide information about the real
and imaginary parts of the bottom-quark coupling to the Higgs boson. Exclusive
decays to $\omega\gamma$ and $K^{*}\gamma$ can probe the flavour-conserving
coupling of the Higgs boson to up and down quarks, and the flavour-violating
coupling of the Higgs boson to down and strange quarks, respectively. The
theoretical predictions for the corresponding branching fractions are tiny and
range from $\mathcal{O}(10^{-6})$ for $J/\psi\gamma$ and $\omega\gamma$ final
states through $\mathcal{O}(10^{-9}-10^{-8})$ for $\Upsilon\gamma$ and a rough
estimate of $\ll 10^{-11}$ for $K^{*}\gamma$. Searches for the above decays
are performed using $\mu^{+}\mu^{-}$ decays of the vector quarkonia [5], while
the decay $\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ is used to reconstruct the
$\omega$ meson, and the decay $K^{*}\rightarrow K^{+}\pi^{-}$ is used to
reconstruct the $K^{*}$ meson [6]. The non-trivial background shape is
estimated using the correlated sampling method [7] and normalised from the fit
to data, as can be seen in Fig. 2 for $J/\psi\gamma$ (a) and $\omega\gamma$
(b).

(a)

(b)

Figure 2: (a) Reconstructed invariant mass $m_{\mu^{+}\mu^{-}\gamma}$ for the
$J\psi\gamma$ [5] and (b) reconstructed invariant mass
$m_{\pi^{+}\pi^{-}\pi^{0}\gamma}$ for the $\omega\gamma$ search [6]. The
branching fraction of each of the signals is set to the observed 95% CL upper
limit.

The observed data are compatible with the expected backgrounds. The 95% CLs
upper limits on the branching fractions of the Higgs boson decays into
$J/\psi\gamma$, $\psi(2S)\gamma$, and $\Upsilon(1S,2S,3S)\gamma$ are found to
be $2.1\times 10^{-4}$, $10.9\times 10^{-4}$, and $(2.6,4.4,3.5)\times
10^{-4}$, respectively, assuming SM production of the Higgs boson. The 95% CL
upper limits on the branching fractions to light mesons and a photon are
$\mathcal{B}(H\rightarrow\omega\gamma)<1.5\times 10^{-4}$ and
$\mathcal{B}(H\rightarrow K^{*}\gamma)<8.9\times 10^{-5}$. An earlier analysis
based on partial Run 2 ATLAS data extracted limits for the Higgs boson decays
into $\rho\gamma$ and $\phi\gamma$ exclusive decays [8].

##  3 Searches for lepton flavour violation in the Higgs sector

Neutrino oscillations prove that lepton flavour is not an exact symmetry of
nature. Higgs sector may be responsible. LFV naturally occurs in more than one
Higgs dublet models, composite Higgs or Randall-Sundrum warped extra
dimensions. The searches reported below allow to put direct limits on the off-
diagonal elements ($Y_{\tau\mu},\ Y_{\tau e},\ Y_{e\mu}$) of the Yukawa
coupling matrix without making any assumptions about diagonal couplings.
Search for the Higgs boson decay to either a $e\tau$ or a $\mu\tau$ pair
The direct search for lepton flavour violation in Higgs boson decays,
$H\rightarrow e\tau$ and $H\rightarrow\mu\tau$ is performed using both the
leptonic and hadronic $\tau$ decay channels [9]. For leptonic tau decays only
combinations with the different flavour lepton are considered. Two background
estimation techniques are employed: the MC-template method, based on data-
corrected simulation samples, and the Symmetry method, based on exploiting the
symmetry between electrons and muons in the SM backgrounds. No significant
excess of events is observed and the results are interpreted as upper limits
on lepton-flavour-violating branching ratios of the Higgs boson. The observed
(expected) upper limits set on the branching ratios at 95% CL,
$\mathcal{B}(H\rightarrow e\tau)<0.20\%\ (0.12\%)$ and
$\mathcal{B}(H\rightarrow\mu\tau)<0.18\%\ (0.09\%)$, are obtained with the MC-
template method from a simultaneous measurement of potential $H\rightarrow
e\tau$ and $H\rightarrow\mu\tau$ signals. The corresponding likelihood contour
is shown in Fig. 3(a), displaying a mild $2.1\sigma$ tension with the SM
expectation. The best-fit branching ratio difference,
$\mathcal{B}(H\rightarrow\mu\tau)-\mathcal{B}(H\rightarrow e\tau)$, measured
with the Symmetry method in the channel where the $\tau$-lepton decays to
leptons, is $(0.25\pm 0.10)\%$, compatible with a value of zero within
$2.5\sigma$.

(a)

(b)

Figure 3:  (a) Best-fit value (red star) of the branching ratios
$\mathcal{B}(H\rightarrow e\tau)$ and $\mathcal{B}(H\rightarrow\mu\tau)$,
given in % , and likelihood contours at 68% and 95% CL obtained from the
simultaneous fit of $H\rightarrow e\tau$ and $H\rightarrow\mu\tau$ signals
based on the MC-template method, compared with the SM expectation (black
cross) [9]. (b) Simultaneous background-only fit to $e\mu$ data in all signal
region cetegories. The signal parameterisation with branching fraction set to
$\mathcal{B}(H\rightarrow e\mu)=0.05\%$ is also shown (red line) [10].

Search for the Higgs boson decay to a $e\mu$ pair
The measurement relies on simultaneous fit of the narrow resonance on top of
the continuum backgrond in the invariant mass distributions reconstructed in
multiple categories of signal regions [10]. The categorisation is based on
production mechanism, centrality and the transverse momentum of the lepton
pair. Main backrounds include Drell-Yan $\tau\tau$ pair production,
$t\bar{t}$, di-boson and misindentified lepton events. The combined result of
the background-only fit to all signal cetegories is shown in Fig. 3(b). No
evidence of the decay $H\rightarrow e\mu$ is observed, with the best-fit value
of the branching fraction of $(0.4\pm 2.9({\rm stat.})\pm 0.3({\rm
syst.}))\times 10^{-5}$. The observed (expected) upper limit at the 95% CL is
found to be $6.2\times 10^{-5}\ \ (5.9\times 10^{-5})$.

(a)

(b)

Figure 4:  (a) Simultaneous background-only fit to $ee$ data in all signal
region cetegories. The signal parameterisation with branching fraction set to
$\mathcal{B}(H\rightarrow ee)=2\%$ is also shown (red line) [10]. (b)
Simultaneous background-only fit to $\mu\mu$ data in all signal region
cetegories weighted by $\log(1+S/B)$. The lower panel compares the fitted
signal pdf, normalised to the signal best-fit value, to the difference between
the data and the background model [11].

##  4 Higgs boson decays to 1st and 2nd generation leptons

SM predicts $\mathcal{B}(H\rightarrow ee)_{\rm SM}\sim 5\times 10^{-9}$ and
$\mathcal{B}(H\rightarrow\mu\mu)_{\rm SM}\sim 2\times 10^{-4}$. Experimental
verification of these predictions is important but challenging. At the same
time, any significant enhancement of the branching fraction would point to
phenomena from beyond the SM.
Search for the Higgs boson decay to a $ee$ pair
The same analyis [10] described above is also utilised to look for Higgs
decays to opositely signed electron pairs. Here, background is substantially
larger and dominated by the Drell-Yan $ee$ production. No evidence of the
decay $H\rightarrow ee$ is observed. The best-fit value of the branching
fraction is $(0.0\pm 1.7({\rm stat.})\pm 0.6({\rm syst.}))\times 10^{-4}$. The
observed (expected) upper limit on the branching fraction at the 95% CL, is
found to be $3.6\times 10^{-4}\ \ (3.5\times 10^{-4})$.

Search for the Higgs boson decay to a $\mu\mu$ pair
A similar analysis is performed in search for muonic Higgs decays [11]. It
defines 20 mutually exclusive signal regions based on the production mechanism
and S/B ratio. The clasification is done by boosted decision tree classifiers
constructed for each of the four production mechanisms (ggF, VBF, VH and ttH).
A fit to the reconstructed invariant mass of the muon pair is performed in all
signal categories simultaneously. The invariant mass distribution weighted sum
in all signal regions together with the fitted signal is shown on Fig. 4(b).
The observed (expected) significance over the background-only hypothesis for a
Higgs boson with a mass of 125.09 GeV is $2.0\sigma\ \ (1.7\sigma)$. The
observed upper limit on the cross section times branching ratio for
$pp\rightarrow H\rightarrow\mu\mu$ is $2.2$ times the SM prediction at 95% CL,
while the expected limit on the $H\rightarrow\mu\mu$ signal assuming the
absence (presence) of a SM signal is $1.1\ \ (2.0)$. The best-fit value of the
signal strength parameter, defined as the ratio of the observed signal yield
to the one expected in the SM, is $\mu=1.2\pm 0.6$.

##  5 Concluding remarks

Searches for strongly suppressed or forbidden decays of the Higgs boson offer
an atractive probe of phenomena from beyond the SM. A wide range of such
searches have been performed by the ATLAS Colllaboration. The full review is
beyond the scope of this contribution. Instead, the reader is encouraged to
consult the ATLAS public results webpage [12]. So far, no significant
deviation from the SM predictions have been observed. The sensitivity of the
Higgs decay to a pair of muons is currently at the virge of an evidence.
Observation of the muonic Higgs decays as well as confirmation or disproving
of the currently seen departures fro SM expectations (e.g. the LFV Higgs decay
in the $\ell\tau$ channel) will require considerably more data to come with
the ongoing LHC Run 3 and subsequently with the high luminosity phase of the
LHC.

Acknowledgements This work is supported in part by the Polish Ministry of
Education and Science project no. 2022/WK/08.

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