LieGG: Studying Learned Lie Group Generators
Authors: Artem Moskalev
Anna Sepliarskaia
Ivan Sosnovik
Arnold W.M. Smeulders
Venue: NeurIPS 2022 Accept (2022)
Keywords: invariance
equivariance
symmetry
Lie groups
interpretability

Abstract: Symmetries built into a neural network have appeared to be very beneficial for a wide range of tasks as it saves the data to learn them. We depart from the position that when symmetries are not built into a model a priori, it is advantageous for robust networks to learn symmetries directly from the data to fit a task function. In this paper, we present a method to extract symmetries learned by a neural network and to evaluate the degree to which a network is invariant to them. With our method, we are able to explicitly retrieve learned invariances in a form of the generators of corresponding Lie-groups without prior knowledge of symmetries in the data. We use the proposed method to study how symmetrical properties depend on a neural network's parameterization and configuration. We found that the ability of a network to learn symmetries generalizes over a range of architectures. However, the quality of learned symmetries depends on the depth and the number of parameters.

LieGG: Studying Learned Lie Group Generators
Artem Moskalev UvA-Bosch Delta Lab University of Amsterdam
Anna Sepliarskaia Machine Learning Research Unit TU Wien seplanna@gmail.. com
Ivan Sosnovik UvA-Bosch Delta Lab University of Amsterdam
sossoovikuuva
Arnold Smeulders UvA-Bosch Delta Lab University of Amsterdam
a W. www.wders@uaa.....mmmmm nl
Abstract
Symmetries built into neural network have appeared be very beneficial for wide range of tasks as it saves the data to learn them. We depart from the position that when symmetries are built into a model priori, is advantageous for robust networks to learn symmetries directly from the data to fit task function. In this paper, we present method extract symmetries learned by neural network and to evaluate the degree to which network is invariant to them. With our method, we are able to explicitly retrieve learned invariances in form of the generators of corresponding Lie-gne Lie-groups groups without prior knowledge of symmetries in the data. We use the proposed method study how symmetrical properties depend on a neural network's 's parameterization and configuration. We found that the ability of a network learn symmetries generalizes over a range architectures. However, the quality of learned symmetries depends on the depth and the number of parameters.
Introduction
Convolutional Neural Networks (CNNs) are efficient, one because they can convert the translation symmetry in the data into built-in caasslatoo-equivariann property of the network without exhaust- ing the data to learn the equivariance. Group-equivariant networks generalize this property to rotation
39 20] scale [35 33 4, and other symmetries defined by matrix groups ([4] Equipping neural network with prior known symmetries has proved to be data-cfficient.
66, 42
466) have demonstrated that hard coding the symmetry into neural network does not always lead to a better generalization. Often, soft or learned equivariance more ad- vantageous mostly in terms of data efficiency and accuracy [38]. Moreover, architectures with no equivariance built-in, such as transformers [IT] and multi-layer perceptron mixers [36], achieve remarkable performance priors without explicit symmetry constraints [ This raises following questions to study this paper. To what degree do neural networks learn symmetries? How accurately do learned symmetries reflect the true symmetrics in the data? Can we support the capability of neural networks for learning symmetries? We present method to study symmetries learned by neural networks address the questions.
Recent works 13 21
wide range of problems. Effectively, they learn their own geometrical
In the paper, we depart from Lie group theory and develop a method that can retrieve symmetries learned from the data by any neural network (Figure [T] Our method only makes the assumption that
Source code: htttps://githeii
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

g exp(b
0
X
VF
F
X
F
Training data
Polartzation Matrix
We solve the matrix nullspace equation to derive infinitesimal generator from neural network and the training data. Due to Lie algebra Lie group correspondence, we can calculate the class of transformations the model is invariant to by exponentiating the generator.
Figure 1:
model is differentiable, commonly met in neural networks. While previous works on analyzing symmetries in neural networks rely empirical analysis of network representations [28] or on examination of a given set transformations [15 33 25), our method outputs a generator of corresponding Lie-group and allows to quantitatively evaluate how sensitive the network is in the direction of the learned symmetry We make the following contributions:
From the perspective of Lie-groups, we propose the theory to study symmetrical properties of neural networks.
We derive an efficient implementation of the method that allows improving the interpretabil- ity of the model by revealing symmetries it learned.
With our method we conclude that models with more parameters and gradual fine-tuning learn more precise symmetry groups with a higher degree of invariance them.
2
Related work
Equivariant and invariant networks The goal of equivariant and invariant networks is to build the symmetries into neural network architecture as inductive bias. Starting from the convolutional networks (24) that contain a translation symmetry, the concept of the equivariance was generalized to
rotations 6
34 41 and arbitrary matrix groups in Various methods have been proposed for learning symmetries directly from the training data
42 39
20, permutations [44, scaling 35
33
14
among which a popular approach is to learn transformations by estimating the infinitesimal generators of symmetry groups [32
46
10 7 31 9. These papers focus on building the symmetries into the models by modifying the architecture. In our work, we focus the reverse question, i.e. given a network with a fixed architecture, what symmetries does it learn from the data?
Symmetries in neural networks Another line of work is focused interpreting symmetries in neural networks. In (28) Olah et al. take inspiration from biological circuit motifs [2] and study cquivariant patterns learned by an unconstrained neural network on the image classification task. The authors demonstrate that the network learns rotation, hue, and scale symmetries when trained on ImageNet. In [15] Goodfellow al. propose a number of empirical tests study invariances to known transformations in network, and demonstrate that auto encoding architectures learn increasingly invariant features in the deeper layers. In [25] Lenc Vedaldi quantify invariances and equivariance in the layers of convolutional networks to pre-defined set of transformations. In the concurrent work of Gruver al. (16), the authors propose using the Lie derivative measure local equivariance error to known symmetry groups. In contrast to these works, our method does not require knowing the set transformations beforehand; and it also provides theoretical guarantees of the invariance from the perspective of the Lie groups theory.
Network configuration: width, depth, number parameters Neural networks of various widths and depths have been studied through the lens of universal approximation theorem (26 22 functional expressiveness 30) and by empirical analysis of learned representations (27]. These works focus on either characterizing learning capabilities of neural networks interpreting the differences

---

between representations that models with various architectures learn. They, however, do not analyze how the symmetrical properties of a model depend on its width and depth.
Other works focus on analyzing a connection between a number of parameters and the generalization capability of networks 4 37 45 It is commonly argued that overparametrized architectures achieve better generalization bounds, i.c. provide smaller discrepancy between train and test perfor- mance and thus preferred in many applications [45). However, it is unclear if the models with more parameters and hence with a finer generalization capability also learn symmetries better. In our paper, we investigate this question for family of feed-fome forward networks.
Robust Learning Various methods have been proposed to train more robust neural networks. While a naive training may lead to overfitting on the training subset, a well-organized pre-training helps to mitigate this issue [2]. Features extracted by a model pretrained on a bigger dataset often demonstrate more robust results [88]. While there are many effective techniques for training more robust feature extractors [5] 17) the analysis of such methods from the invariance point of view has not been done before. We demonstrate that our method allows for better understanding and explaining what training regimes lead to more invariant, and thus more robust representations.
3 Background
The focus of this paper is a symmetry group. A symmetry a transformation of an object that leaves the object unchanged. The set of all such transformations of an object with composition as binary operation forms a group. In this paper, the object of interest a dataset and its symmetries. We study what kind of transformations map data sample to another and how neural networks learn information about the symmetries.
The theory of Lie groups is sweet spot of mathematics that helps to formalize symmetries and provides practical methods for studying them. Formally, a Lie group is a group that has the structure of differential manifold. The tangent space in the identity element of Lie group forms vector space called Lie algebra. Lie algebra determines Lie group up to an isomorphism for simply connected Lie groups and, being vector space, is more convenient object to study than Lie group. Thus, in order to recover a simply connected symmetry group, it is sufficient to understand its Lie algebra.
In this paper, we assume that the Lie group G is a matrix Lie group, i.e. is closed subgroup of the general linear group of the degree n, denoted as GT(n). It is defined the set of n x n invertible matrices. The correspondence between the Lie group G and Lie algebra g in this case given by the exponential map:
g = = M() eh E GYYERR)
where (n) is the of all n matrices. Each such element h from equation I is called an infinitesimal generator of the Lie group G.
We introduce LieGG, a method to compute Lie Group Generators. LieGG extracts the infinitesimal generators of the Lie algebra for symmetries that a neural network learned from the data. To find the Lie algebra basis,
the discriminator of the dataset, i.e. function F : R - R such that F(x) = 0 if and only if x is an element of the dataset D. The discriminator function is naturally modeled by neural network fitting the data subject downstream task. With this, we use the following criterion for symmetry group:
Theorem 3.1 (Theorem 2.8 (29). Let G be a connected Lie group of linear transformations acting on an n-th dimensional manifold X. Let F X - R,, IS n, define system of algebraic equations:
Fx(x) 0, = 1
and assume that the system is of maximal rank, meaning that the Jacobian matrix for every solution x of the system. Then is symmetry group of the system if and only if
(8Fv
is of rank I
3

---

i=njz=n
OFv.
Σ
3x; hij Iy 0, whenever Fu(x)
1,5=1
for 1,... ,I and every infinitesimal generator h of G. where bij is an element of the matrix h in the i-th row and j-th column.
Example We illustrate the practical significance of the theorem with example. Suppose the dataset D lies on a sphere in I::
D= fx EXER" xf
2 1).
The discriminator for this example the function F(x) = x
According to the theorem, to find the Lie algebra of the symmetry group, we can find a basis of the solution of the following linear equations, where az are the variables interest:
Σ F3 dij Dj =0
when 2² + ... + x = 1. We assert that the solutions are the family of matrices A such that A + A" = 0. Indeed, for H; 1, 2j 0, j i, the condition means that as 0. For 1
2; V2 Ij / Ik 0, o i,j. it follows that aij + azi 0. On the other hand,
A = - AT implies Eie (z: 0zj 2j xy ags- x.) = Enj (z+ asj 2j
agj 23) 0.
Note that the Lie algebra of the matrices (A: A+ A 0} corresponds to the Lie group of matrices {B: B B = }} which is the rotation group. Thus, the symmetry of a sphere is the rotation group.
Method
To compute Lie algebra given discriminator function F R -> the system of linear equations, where each equation corresponds to one element in the dataset:
R Theorem 3.1 and solve
i=nj=n OF
bis 2j = 0, for each point in the dataset.
(2)
Oz:
This is a system of linear equations with n² variables bij and the number of equations equal to the number of points in the dataset. We can cast solving such system with respect finding nullspace of the matrix E. The matrix & has the number of rows equal to the number of points in the dataset and n columns. Multiplying & with the vectorized representation of yields the system of equations in 2 We call the matrix & the network polarization matrix.
problem of
The problem of finding the nullspace basis of matrix is standard problem in the numerical analysis. can be efficiently solved using the Singular Value Decomposition (SVD). Recall that we can write matrix using the SVD in the following way UEVT, where U, V are orthogonal matrices and a diagonal matrix with decreasing singular values. Thus, the columns in V corresponding to nearly-zero singular values encode the nullspace of the system of equations in 2 and hence form Lie algebra basis.
Thereby, in practice, the calculation of LieGG consists of 3 steps: (i) training a neural network, (ii) calculation of the polarization matrix E, (iii) computing singular vectors corresponding to almost zero singular values of the polarization matrix E.
4.1 Computing a Lie algebra of group acting on R22
Symmetries play an important role computer vision problems, and we will describe LieGG for this case in more details. In various computer vision problems, it is assumed that the symmetry group G changes images from the dataset by acting on R2. For example, convolutional networks (44 and recently proposed equivariant networks for rotations 42 [9] assume that the group acts

---

translation or rotation of an image. These neural networks perceive each image as a function ff with non-zero values on some compact subset ! C R²2. where the elements of the subset correspond to spatial coordinates. Thus, the space of images I is the space of all functions that map spatial coordinated to pixel values. In other words: I == (ʃ, f Ω + R.. A vision model F I + R is function on such space of images.
We view an image is function on pixel grid (x1,-... Xn) of the domain Ω2 i.e. the image can be identified with the point in R" which encodes the pixel values. In this case the model takes n real values as an input: ((((x)).... f(xx))
important to note that Theorem 3.1 assumes that the group G acts continuously on the data manifold. However, this assumption violated for the image data an image is non-continuous function of coordinates due to the discrete pixel grid. We overcome the discontinuity issue by preprocessing images with Gaussian smoothing.
To use LieGG for the image data, we explicitly write the condition that the matrix is an infinitesimal generator for the image data I on the grid (x1,"-",x,, The condition reads as follows:
p=n,k =
OF
a5(x)) Ozb
F(x() .. f(e o xx,))= = =0 ++
Σ 1=1,==1== 0x(xp)
haj m =0 (3)
where E R and ny corresponds the projection of x, onto the j-th coordinate axis. The resulting condition is the equation in 4 variables baj. k = 1, j 1, Hence, the polarization matrix has columns. Thus, in equation a space of potential image symmetries is reduced to a 4-dimensional spatial subgroup. This is to provide prior for LieGG to retrieve meaningful symmetries from for images as the full-scale space of potential symmetries overwhelming for the image data, This is
strong yet proven to useful prior for vision models 66 19
39 20)
In practice, the caleulation of LieGG for vision models consists of the following steps: (i) prepro- cessing images by doing Gaussian smoothing, (ii) training a neural network, (iii) computing the polarization matrix using equation 3 (iv) computing singular vectors corresponding to almost zero singular values of the polarization matrix.
4.2 Symmetry bias and symmetry variance
The scope of LieGG is not limited to retrieving the generator of a learned symmetry group. With the polarization matrix ε, we can also evaluate the degree of the neural network's 's invariance to learned symmetry. Furthermore, if true symmetry in the data is known, we can apply our method to analyze how close is a learned symmetry to the true one. The former criterion we alias symmelry variance, and the later- symmetry bias.
Firstly, we show how to estimate the degree of the network invariance to the element of a Lic group:
R(g) = E(F F(x))2 ~ D
Σ (Flg xx) F(xx)) = R(g))
X.ED
Let h be an element from Lie algebra that corresponds to i.e.
R(g) = Σ
1
(F(exp (ht) x;) F(x;))?
D
x:ED
aF(x))
hki + +(+)(1))2
D
Oz
x
kjj
(F(x)
Σ
2
haj x1)2
dxf
X.ED
k.j
5

---

The matrix with the i-th row equal to @F(x)) dx x! equates to the polarization matrix &. Then
R(g) = DJ (1²
2 || 1 O
where h is the vectorized representation of the element of the Lie algebra. To calculate the value of Eh we use the SVD of the matrix & Let UEVT, where U, are semi-unitary matrices and a diagonal matrix. With this, we can estimate R(g) in the following way.
Σ (V R(g) ~ |J 11 DDI
1
(4)
where oj is the 1-th singular value of the network polarization matrix &
In particular, the equation reveals that singular values of the network polarization matrix reflect the degree of invariance to transformations encoded by the corresponding singular vectors. With this, use the smallest singular value of the network polarization matrix to define the synmetry variance as V(E) min (E)²/ DD] Symmetry variance estimates the degree of invariance in the direction of the symmetry, network learned to be most invariant to. Moreover, when the true symmetries are known, the difference between the singular vectors and the generator of the true symmetry group equates to the symmetry bias, i.e. (e 9) = ((: Gl[ where (E(E) returns the i-th singular vector reshaped to matrix, and 9 is ground truth symmetry group generator closest to the (E), e.g. the closest kkew-symmetri matrix for the rotation group.
Note that the proposed symmetry variance allows measuring learned invariance without the knowledge of a symmetry generator beforehand. This is in contrast to Gruver et al. [6]. where the knowledge of symmetry generator is required to measure learned equivariance.
5
Experiments
5.1 Retrieving symmetries from the synthetic regression
Firstly, conduct the experiment the synthetic problem test the ability of method to accurately retrieve a symmetry learned by a network. To do so, we adopt the invariant regression task from [14 with clean 0(5) symmetry built-in. The regression function f R2x5 R is given by:
f(x1, x2) = sin[|xx 0.5 -11
X2
X2 lz
With this, we construct the training dataset to fit the regression function with the multi-layer percep- tron. Here the MLP is the unconstrained feed-forward neural network with prior knowledge of symmetries in the data. Once the model is converged, we apply our method to analyze how close
Polarization spectrum
0(5) symmetry bias
Mean symmetry variance
1.00
0.75
10:
0.50
101
0:25
10
10
0.00
i-th singular value
i-th singular vector
Training set size
Figure 2: Singular values of the network polarization matrix and the symmetry biases per singular vector of the multi-layer perceptron fitted on 0(5) invariant regression task. (right) Comparing the mean symmetry variance for various training set sizes.

---

Polarization spectrum
9-202.5
10"2
10
S012) symmetry bias
10"
4
deep
10
shalloo
i-th singular value
(a)
(b)
Figure 3: (a) Symmetrical properties of the shallow and deep networks: singular values of the network polarization matrix and SO(2) symmetry bias. (b) Shaded: the kernel density of learned 2D rotations on perfect circle. Non shaded: visualizing learned Lie algebras for various rotation angles (in columns). Top row: the shallow network. Bottom row: the deeper network.
are retrieved symmetries clean 0(5) group (symmetry bias), and how invariant is the network to the learned symmetries (symmetry variance). We also study how the ability of the network to learn symmetries depends on the number of available training samples. We report the results in Figure 2
As can be seen from Figure the spectrum of the network polarization matrix is zeroing, indicating that the model becomes invariant the transformations encoded by the corresponding singular vectors. Also, these singular vectors are the generators of the rotation group as indicated by zero symmetry bias with respect ((5) group (Figure (2)). We also observed that the network accurately learns the symmetry group generators even with a small training set size. However, larger training set size leads smaller symmetry variance, i.e. a higher degree of invariance. For an additional validation, we used LieGG to extract symmetries from the randomly initialized 0(5) invariant EMLP 14 model. For this case, LieGG recorded zero symmetry variance and zero symmetry bias with respect to O(5) group without any model pre-training.
The experiment demonstrates that, on the synthetic regression task, our method accurately retrieve symmetries learned or built-in. When symmetries are learned, the larger training dataset facilitates learning invariances for the model.
5.2 Retrieving symmetries from the rotation MNIST
In this experiment, we test the ability of our method to retrieve learned symmetries on the rotation MNIST dataset (23). We train the feed-forward network to output the class of a digit invariant to 2D in-plane rotations. The symmetry group that the model should learn is thus SO(2). We note that different from the synthetic regression task, the symmetries in the rotation MNIST are much noisier since the rotations are coarsened by the projection onto a pixel grid. In addition, the network has classification error, hence it serves as only an approximate data discriminator function. In this scenario, we test how well our method can retrieve noisy SO(2) symmetry.
We experiment with two architectures: the shallow 2-layer perceptron and the deeper 6-layer network with times number of parameters. Once the network's performance plateaus on the validation split, terminate the training, and apply method to retrieve learned symmetries. We plot the spectrum of the network polarization matrix, SO(2) symmetry bias and visualize learned symmetries for the shallow and deep models in Figure 3
From Figure D we observe that for both shallow and deep models, our method retrieves the rotation symmetry indicated by the zeroing symmetry bias with respect to SO(2) group. Also, we noticed that the symmetry Iearned by the shallow model noisier than the symmetry learned by the deeper network as indicated by the higher symmetry bias. We visualize this in Figure 3p by sampling learned Lie algebras and applying them to transform an input image. We see that the symmetry learned by the shallow model diverges from clean rotation and also contains scaling and shearing. We also see, that on the rotation-MNeN the spectrum the network polarization matrix is zeroing not as fast as
7

---

symmetry variance
50(2) symmetry bias
0.35
1--
40000
80000
symmettry valiaal lelaance
0.5
16-2
p= -0.49
symmetry blas
p=
-0.31
0.25
0.12
160000
0.3
0.2
0.35
0.07
0.3
0.05
0.75
0.80
0.55
0.0
0.75
0.80
0.85
Accuracy
hidden layers
(a)
(b)
Figure 4: (a) Symmetry variance and SO(2) symmetry bias for the networks with different width, depth and number of parameters. (b) The correlation plot the symmetry variance and the symmetry bias versus the test accuracy of the models. Each point corresponds to one network. The color indicates the number parameters.
in the synthetic case. It indicates that the models, although do learn the rotation symmetry, do not become fully invariant it.
This experiment demonstrates that our method can retrieve symmetries from the data even when an imprecise discriminator function is used.
5.3 Symmetries in networks with different configurations
Observing the differences in symmetrical properties for different network configurations in Experi- ment 5.2] we further conduct the study how the symmetry variance and the symmetry bias depend on width, depth, and number parameters in a neural network. In this experiment, we are interested network configurations it is easier to learn symmetries from the data.
to know if for
To obtain different configurations, we sample number parameters from the range of [40000, 200000] with the step size of 20000, and vary the depth of a network from 1 to 5 hid- den layers. We train all the networks until convergence on the validation split and apply LieGG to retrieve the symmetries and to record the symmetry variance and the symmetry bias. We also track the accuracy of the models to correlate it with the symmetry variance and the symmetry bias.
The results are reported in Figure We summarize the analysis of how the configurations of the network influence its symmetrical properties:
Number of parameters: we observe that the networks with more parameters tend to learn to the higher degree of invariance than their shallow counterparts indicated by consistently smaller sym- metry variance. Also, we see that the networks with fewer parameters learn less precise symmetries as indicated by the higher symmetry bias.
Depth / width: our results indicate that sufficiently parameterized deep networks are more capable to learn invariances than the wide ones with the same number parameters. In other words, deeper networks become more invariant in the direction of learned symmetries. At the same time, the same only partially holds for the quality of learned symmetry group as indicated by the fluctuating symmetry bias, when the depth 3. Overall, the network with balanced width and depth achieves the smallest symmetry bias.
we also test how the symmetry variance and the symmetry bias correlate with the performance of the models. As can seen from Figure D the symmetry variance and bias have moderate negative correlation coefficients Pvar -0.49 (p-value < 1e and Pbias -0.31 (p-value < le 4) with the classification accuracy respectively. The trend indicates that more accurate models tend have lower symmetry variance and lower symmetry bias. However, the correlation not strict. That is, is possible to have neural networks, which learn symmetries differently, but perform equally well during the test.
Symmetry vs. accuracy:

---

symmetry variance
50(2) symmetry bias
0.20
MLP-7
sub MLP-3
0.004
0.15
MLP-7 [F]
MIP-7 (L]
0.003
0.10
0.002
0.05
Index of the layer
Figure 5: Symmetry variance and SO(2) symmetry bias measured layer-wise for the network trained on the rotation-MNIS, dataset. Different colors corresoond different training regimes of the model. Training scenarios are: ML.P- for the 7-layer perceptron, sub ML.P-3 for the 3-layer sub-network, MLP--[[F] for the joint fine-tuning, MLP-7[L] for the layer-wise training.
5.4 Tuning layer-wise symmetries deep networks
In the previous experiment, analyzed how the symmetrical properties of the networks depend on the configuration of the model. Now, we ask if by modifying a training regime, we can facilitate learning symmetries for the network. In this experiment, we evaluate two known approaches to robust training: (i) pre-training with fine-tuning, and (ii) layer-wise training (). We also evaluate plain end-to-end training baseline.
To simulate the pre-training of part of the network weights, we take randomly initialized 7 layers perceptron (mother network) and extract the sub-network that consists of the first 3 layers. We then attach the classification head to the sub-network and train it independently from the mother network on the rotation-MNIST dataset. After that, we re-inject the learned weights into the mother network and retrain the model. For the fine-tuning, we train all of the weights jointly. For the layer-wise training, we freeze the pre-trained part and only train the remaining layers. We then apply our method to study the symmetrical properties of the models trained with each of the regimes. We plot the symmetry variance and the symmetry bias after each layer of the network in Figure 5
Final and pen-ultimate layer symmetries: from Figure 5 we observe that for the different training regimes, the network learns approximately the same symmetries at the same degree of invariance in the final layer. Surprisingly, we also noticed that in the pen-ultimate layer, the symmetries for all of the training regimes come very close measured by the symmetry variance and bias. Since the last layers are directly responsible for providing outputs for the downstream task, we hypothesize that there exist preferred optimal symmetry level for neural network that depends and the data, but not from training regime of a model.
architecture
the other hand, observed that symmetries in the intermediate layers of the network do depend on the training regime. Firstly, the pre-trained sub-network learns more accurate symmetry with the higher level of invariance compared to when it is trained as part of the deeper model. We attribute it to the fact that symmetries are defined on a level of training labels, hence invariances learned by the model in the final layer should be kept a minimum [40]. Secondly, after the fine-tuning or the layer-wise training, symmetries in the intermediate layers of the network both more precise and utilized at the higher degree of invariance as indicated by the smaller symmetry bias and variance respectively. It indicates that we can facilitate learning intermediate invariances in a neural network by using a pre-training followed by the fine-tuning or the layer-wise learning approaches. This can be useful for transfer learning when it is important to transfer as much inductive bias as possible into other models to other tasks.
Intermediate symmetries:

---

6 Conclusion
Summary We present LieGG the approach to retrieve symmetries learned by neural networks. Conveniently, our method extracts symmetries directly in the form of corresponding Lie group generators. This allows to explicitly obtain symmetries learned by a neural network without specifying any set of transformations of interest beforehand. We next introduce the symmetry variance and the symmetry bias to evaluate symmetrical properties of neural networks by analyzing the network polarization matrix computed by LieGG. We make use of our approach by studying how the quality of the symmetry learned by the neural network depends on a network's configuration and a training regime. By improving the interpretability of symmetry learning, we identify configurations and training regimes that facilitate training invariances in neural networks.
Limitations and Future Work The proposed method can facilitate the interpretability of neural networks and help to design efficient ways of incorporating symmetries into a model. However, some limitations remain. In particular, LieGG is currently limited to instances of GL(n) connected group, which hinders the analysis of complex invariances, c.g. where a transformation can not be described group action. It is an intriguing direction for future research to extend a mechanism of retrieving symmetries to more complex transformations. Nonetheless, LieGG provides an important building block on the way increasing the interpretability of symmetry learning in neural networks. Also, in our analysis, we focus on the specific family of models, i.e. feed-forward multi-layer perceptions trained on the datasets with well-controlled factors of variations. We assume that further large-scale analysis of LieGG in complex scenarios and with more advanced models can provide a more general understanding of symmetry learning in neural networks.
7
Acknowledgments and Funding Disclosure
We would like to thank Evgenii Egorov and Volker Fischer for their valuable comments and discus- sions. We thank the reviewers for their insightful feedback and helpful suggestions that improved our paper. This work has been financially supported by Bosch, the University of Amsterdam, and the allowance of Top consortia for Knowledge and Innovation (TKIs) from the Netherlands Ministry of Economic Affairs and Climate Policy.
References
[1] Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparam- eterized neural networks, going beyond two layers. In Proceedings of the 33rd International Conference on Neural Information Processing Systems (NeurIPS), 2019.
[2] Uri Alon. An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman and Hall/CRC, 2006.
[3] Fabio Anselmi, Georgios Evangelopoulos, Lorenzo Rosasco, and Tomaso Poggio. Symmetry- adapted representation learning. Pattern Recognition, 86:201-208, 2019.
[4] Erik Bekkers. B-spline cnns on lie groups. In International Conference on Learning Repre- sentations (ICLR), 2020.
[5] Y. Bengio, Pascal Lamblin, Dan Popovici, Hugo Larochelle, and U. Montreal, Greedy layer-wise training of deep networks. In Advances in Neural Information Processing Systems, 2007.
[6] Taco Cohen and Max Welling, Group equivariant convolutional networks. In Proceedings of The 33rd International Conference on Machine Learning (ICML), 2016.
[7] Marissa Connor and Christopher Rozell. Representing closed transformation paths in en- coded network latent space. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, 2020.
[8] -aptisee Cordonnier, Andreas Loukas, and Martin Jaggi. On the relationship between self- attention and convolutional layers. In International Conference on Learning Representations (ICLR), 2020.
[9] Benjamin Culpepper and Bruno Olshausen. Learning transport operators for image manifolds. Advances in neural information processing systems, 22, 2009.
10

---

[10] Nima Dehmamy, Robin Walters, Yanchen Liu, Dashun Wang, and Rose Yu. Automatic symmetry discovery with lie algebra convolutional network. In Advances in Neural Information Processing Systems (NeurIPS), 2021.
[11]
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. International Conference on Learning Representations (ICLR), 2021.
[12] Dumitru Erhan, Aaron Courville, Yoshua Bengio, and Pascal Vincent. Why does unsupervised pre-training help deep learning? In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 201-208. JMLR Workshop and Conference Proceedings, 2010.
[13] Mare Finzi, Gregory Benton, and Andrew G Wilson. Residual pathway priors for soft equivari- ance constraints. In Advances in Neural Information Processing Systems (NeurIPS), 2021.
[14] Mare Finzi, Max Welling, and Andrew Gordon Wilson. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. In International Conference on Machine Learning (ICML), 2021.
[15] Ian Goodfellow, Honglak Lee, Quoc Le, Andrew Saxe, and Andrew Ng. Measuring invariances in deep networks. In Advances in Neural Information Processing Systems (NeurIPS), 2009.
[16] Nate Gruver, Marc Finzi, Micah Goldblum, and Andrew Gordon Wilson. The lie derivative for measuring learned equivariance, 2022.
[17] Kaiming He, Ross Girshick, and Piotr Dollár. Rethinking imagenet pre-training. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019.
[18] Dan Hendrycks, Kimin Lee, and Mantas Mazeika. Using pre-training can improve model robustness and uncertainty. In International Conference on Machine Learning (ICML), 2019.
[19] J-H. Jacobsen, B. de Brabandere, and A. W.M. M. Smeulders. Dynamic steerable blocks in deep residual networks. In British Machine Vision Conference (BMVC), 2017.
[20] Erik Jenner and Maurice Weiler. Steerable partial differential operators for equivariant neural networks. In International Conference on Learning Representations (ICLR), 2022.
[21] Naman Khetan, Tushar Arora, Samee Ur Rehman, and Deepak K Gupta. Implicit equivariance
in convolutional networks. arXiv preprint arXiv:2111.14157. 2021.
[22] Patrick Kidger and Terry Lyons. Universal Approximation with Deep Narrow Networks. In Proceedings of Thirty Third Conference on Learning Theory, volume 125 of Proceedings of Machine Learning Research, pages 2306-2321 2327. PMLR, 09-12 Jul 2020.
[23] Hugo Larochelle, Dumitru Erhan, Aaron Courville, James Bergstra, and Yoshua Bengio. An empirical evaluation of deep architectures on problems with many factors of variation. In Proceedings of the 24th International Conference Machine Learning (ICML), 2007.
[24] Yann LeCun, Patrick Haffner, Léon Bottou, and Yoshua Bengio. Object recognition with gradient-based learning. In Shape, contour and grouping in computer vision. Springer, 1999.
[25] Karel Lenc and Andrea Vedaldi. Understanding image representations by measuring their equivariance and equivalence. In Proceedings of the IEEE conference on computer vision and pattem recognition (CVPR), 2015.
[26] Zhou Lu, Hongming Pu, Feicheng Wang, Zhiqiang Hu, and Liwei Wang. The expressive power of neural networks: A view from the width. Advances in Neural Information Processing Systems (NeurIPS), 30, 2017.
[27] Thao Nguyen. Maithra Raghu, and Simon Kornblith. Do wide and deep networks learn the same things? uncovering how neural network representations vary with width and depth. In International Conference on Learning Representations (ICML), 2021.
[28] Chris Olah, Nick Cammarata, Chelsea Voss, Ludwig Schubert, and Gabriel Goh. Naturally occur- ring equivariance in neural networks. Distill, 2020. https:/diitilllpuu//0000 pubb/00200cieeuixteuuaaaaaaa
[29] Peter J Olver. Applications of Lie groups to differential equations, volume 107. Springer Science & Business Media, 2000.

---

[30] Maithra Raghu, Ben Poole, Jon Kleinberg, Surya Ganguli, and Jascha Sohl-Dickstein. On the expressive power of deep neural networks. In Proceedings of the 34th International Conference on Machine Learning (ICML., 2017.
[31] Rajesh Rao and Daniel Ruderman. Learning lie groups for invariant visual perception. Advances in neural information processing systems, 11, 1998.
[32] Jascha Sohl-Dickstein, Ching Ming Wang, and Bruno A Olshausen. An unsupervised algorithm for learning lie group transformations. arXiv preprint arXiv:1001.1027 2010.
[33] Ivan Sosnovik, Artem Moskalev, and Arnold Smeulders. Disco: accurate discrete scale convo- lutions. In British Machine Vision Conference (BMVC), 2021.
[34] Ivan Sosnovik, Artem Moskalev, and Arnold Smeulders. How to transform kernels for scale- convolutions. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) Workshops, 2021.
[35] Ivan Sosnovik, Michal Szmaja, and Amold Smeulders. Scale-cquivariant steerable networks. In International Conference Learning Representations (ICLR), 2020.
[36] Hya O Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Unterthiner, Jessica Yung, Andreas Steiner, Daniel Keysers, Jakob Uszkoreit, et al. MIp- mixer: An mll-mlp architecture for vision. Advances in Neural Information Processing Systems (NeurIPS), 2021.
[37] Sattar Vakili, Michael Bromberg, Jezabel Garcia, Da-shan Shiu, and Alberto Bernacchia. Uniform generalization bounds for overparameterized neural networks. arXiv preprint arXiv:2/09.06099 2021.
[38] Rui Wang, Robin Walters, and Rose Yu. Approximately equivariant networks for imperfectly symmetric dynamics. In International Conference on Machine Learning (ICML), 2022.
[39] Maurice Weiler and Gabriele Cesa. General Equi-Eaqi Stecrable CNNs. Conference on Neural Information Processing Systems (NeurIPS), 2019.
[40] Stephan Wojtowytsch and E Weinan. On the emergence of tetrahedral symmetry in the final and penultimate layers of neural network classifiers. arXiv preprint urXiv:2012.054200 2020.
[41] Daniel Worrall and Max Welling. Deep scale-spaces: Equivariance over scale. Advances in Neural Information Processing Systems (NewrIPS), 2019.
[42] Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov. and Gabriel J. Brostow. Harmonic networks: Deep translation and rotation equivariance. 2017 IEEE Conference Computer Vision and Pattern Recognition (CVPR), pages 7168-7177, 2017.
[43] Dmitry Yarotsky. Error bounds for approximations with deep relu networks. Neural Networks, 94:103-114, 2017.
[44] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan Salakhutdinov, and Alexander Smola. Deep sets, 2017.
[45] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning (still) requires rethinking generalization. Communications of the ACM, 64(3):107- 115, 2021.
[46] Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. In International Conference on Learning Representations (ICLR), 2020.
[47] Barret Zoph, Golnaz Ghiasi, Tsung-Yi Yi Lin, Yin Cui, Hanxiao Liu, Ekin Dogus Cubuk, and Quoc Le. Rethinking pre-training and self-training, Advances in neural information processing systems (NeurIPS), 2020.
12


Reviews:
[{"review_id": "fCW68enNoLt", "reviewer": "Reviewer_erES", "summary": "This paper proposes learned Lie Group Generators (LieGGs) to investigate symmetries encoded in neural networks inversely. It is based on the Lie group-Lie algebra correspondence, which describes an arbitrary matrix Lie group as an exponential map of the corresponding Lie algebra. The authors exploit the fact that infinitesimal generators of a Lie algebra $\\mathfrak{g}$ of a matrix Lie group $G$ acts on data $D$ should satisfy a certain algebraic equation (Theorem 3.1) when there is a $G$-invariant mapping function $F: X \\to \\mathbb{R}$ that satisfies $F(x) = 0$ where $x \\in D$. Because a neural network is an approximation of such $F$ (if the underlying data is $G$-invariant and the neural network learns the data well), the learned symmetries encoded in neural networks can be inversely retrieved by using the proposed method. The authors validate their proposed approach for a simple synthetic problem and rotated MNIST. Also, the authors formulate two metrics that can measure a degree of encoded symmetries, i.e., symmetry bias and variance, based on their approach.", "questions": "As I mentioned, I am generally satisfied with the paper. Some minor comments are the followings:\n\n1. Can the authors elaborate on the effect of the Gaussian smoothing, e.g., is it critical when using the binary rotated MNIST directly?\n2. In (2), isn't $i = j=n$?\n3. In (3), it seems that the underlying group structure is assumed to be a one-parameter sub-group $\\exp(t \\mathfrak{h})$ parameterized by $t \\in \\mathbb{R}$ which is not fully general. Please elaborate on whether the proposed method is applicable for a general $GL(n)$ or its one-parameter subgroup only.\n4. Assume a subset of the rotated MNIST consists of '6's and '9's. '6's and '9's yield identical $G$-orbits while they are not G-invariant with respect to the labeling distribution. Can the proposed method work well for such a subset?\n4. Can the proposed method be used for investigating $G$-equivariance?", "limitations": "The authors state the limitation (limited to $GL(n)$ group) of the proposed method in the section Conclusion. I think that this is an acceptable limitation considering $GL(n)$ is a major and interesting group structure for various problems.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "[Strengths]\n\nExtracting as well as learning symmetries is a definitely crucial problem in the modern deep learning field. While most of the previous works focus on the latter problem, this paper addresses the former one based on a theoretically convincing method, namely Lie algebra. The paper is well-written and the presentation is easy-to-follow. The proposed symmetry bias and variance might be potentially useful metrics to measure the degree of symmetries when building a new $G$-invariant model.\n\n[Weaknesses]\n\nI did not find any critical drawback in this paper. Possible weaknesses include:\n\nThere is no comparison with competitors. Especially, a comparison with the following paper might be important:\n\nN. Dehmamy, R. Walters, Y. Liu, D. Wang, and R. Yu. Automatic Symmetry Discovery with Lie Algebra Convolutional Network. NeurIPS 2021.\n\nDehmamy et al. also use the concept of Lie algebra to deal with arbitrary group symmetries, and as its title indicates, it seems their method might be used for the recovery of underlying symmetries encoded (if I am wrong please correct me). I think the lack of the comparison is not a critical issue because the proposed method is more general, i.e., can be applied to arbitrary neural network architectures - but at least a simple comparison with Lie algebra CNN for the rotation MNIST seems to need for the completeness of the work.\n\nRotated MNIST is only a dataset benchmarked in this paper. Because both MNIST and $SO(2)$ are under relatively simple settings, I am not fully convinced whether the proposed method works well for other datasets or group structures. Possible benchmark candidates might include more complicated images with $SE(2)$.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "X3W1LGcMpL", "reviewer": "Reviewer_BrAS", "summary": "The authors propose LieGG, a method for recovering the Lie group generators corresponding to the symmetries of a trained network. Using their approach, the authors can measure how close a recovered generator is to the true symmetry and to what degree a network is invariant to that transformation. The authors evaluate on synthetic data and MNIST using MLP networks of different depth with known symmetries in the data.", "questions": "* What is v_p in equation 3? \n* How does LieGG perform without pre-processing with Gaussian smoothing? Is spatial smoothness of the input data a requirement for LieGG to perform well? This ablation would be nice to have.\n* What is the p-value corresponding to the correlations in Figure 4b? Please report these in the manuscript.\n* How does this approach scale with dataset size (computational complexity of SVD + sample complexity of LieGG)? It would be nice to include experiments that measure these properties.\n* Is it possible use LieGG to co-train with network weights to decrease the symmetry variance of a network? For example, (Connor and Rozell 2020) show that after a fine-tuning step with learned Lie group generators, the corresponding Lie group provides a significantly better model of MNIST digit rotation.", "limitations": "* The authors metric for symmetry variance seems to fall off in performance as the depth of a network is increased. Perhaps this can be attributed to the approximation to the matrix exponential employed on line 163. The weakness of this approximation has been noted in the literature before in (Rao et al. 1999) vs (Culpepper and Olshausen 2009).", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n* The authors propose a theoretically motivated algorithm for recovering a Lie group generator from a deep neural network that is novel to the best of my knowledge. They evaluate the efficacy of their algorithm on MNIST rotation to show that a deep network trained on rotated MNIST digits implicitly becomes invariant to rotation.\n* The authors propose valuable metrics to evaluate the quality of a learned Lie group and to measure how invariant a model is to a given Lie group.\n\nWeaknesses\n* The authors argue that networks trained with “hard [coded] symmetry” may “not always lead to a better generalization” on lines 20-21, but they never support this claim with experimental data. It would be nice for the authors to compare using their proposed metrics between deep networks that implicitly learn symmetry and those that explicitly incorporate symmetry, such as (Finzi et al. 2021).\n* The experimental results on the invariance of MLPs at different depths are of moderate significance. I would have hoped the authors test their method with larger models and datasets with more advanced invariances. For example, it would be interesting to apply an augmentation to samples from a dataset like CIFAR10 and use LieGG to measure whether the deep network becomes invariant to that augmentation.\n\nMisc fixes:\n* There are several spelling and grammatical mistakes in the experiments section. Please re-read and revise. Here are a few:\n    * “We the MLP” line 178\n    * “closes” line 182\n    * “How the ability of the network to learn” line 193\n    * “The symmetry learn by the” line 220\n* The authors should consider references to a part of the literature on learning Lie group generators from data & incorporating them into networks to encourage invariance (a different, yet related task to recovering Lie group generators from a network to evaluate invariance), such as (\"Learning Lie Groups for Invariant Visual Perception,” Rao et al. 1999), (“Learning transport operators for image manifold,” Culpepper and Olshausen 2009), (“An Unsupervised Algorithm For Learning Lie Groups,” Sohl-Dickstein et al. 2010), and (“Representing Closed Transformation Paths in Encoded Network Latent Space,” Connor and Rozell 2020).\n* Please explicitly state which training scenario corresponds with each entry in the legend of Figure 5.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "LcX8ao_1x5c", "reviewer": "Reviewer_Pv3i", "summary": "The paper presents a method, LieGG, for identifying learned in variances in a trained model.  Specifically, by differentiating the model with respect to its inputs and looking at these derivatives at various inputs points, a polarization matrix is constructed which has as its nullspace, generators of a Lie algebra of invariances. This can be learned using SVD.  The authors apply their method to MLPs trained on a sythentic function with O(5) symmetry and Rotated MNIST which has SO(2) symmetry.  In both cases, the learned model displays the expected invariance.  The authors also analyze the effect of network size, shape, and trained on learned symmetry. ", "questions": "\n## Minor Points\n\n- Line 93: Technically, it is not enough to know the Lie algebra to know the symmetry group.  For example, we discard any discrete symmetries.\n- Line 114: 2 is missing from the equation\n- Line 126: call it _the_ network ...\n- Equation 3 after Line 150: There is no sum over images only over pixel locations.  Should there be a sum over images f as well?\n- Equations after Line 163: How is t sampled?  \n- Equations after Line 163: In the second line of the equation, I think it should be t^2\n- Equations after Line 163: In the third line of the equation, I think there is a missing parenthesis. \n- Equation after Line 164: Should it be O(t^4)?  \n- Line 165: Why is this only an estimate?\n- Equation after Line 167: Should h_l be squared? \n- Line 170: \"Last\" singular values is imprecise. \n- Line 182: with a _closest_ \n- Line 188: the input space is $\\mathbb{R}^{10}$ or $\\mathbb{R}^{2 \\times 5}$. It doesn't really make sense to use both multiplicative and exponential notation for the same thing.  \n- Figure 5: describe abbreviations in caption \n\n## Post-Response Edit\n\nI appreciate the authors response to my questions and maintain my positive score.", "limitations": "The limitations section is good, but the authors could be clearer about what \"more complex transformations\" means.  The assumptions in Theorem 3 and resulting limitations should also be clarified. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strength\n\n- This work is very timely and the question is important.  It is becoming increasingly clear that symmetry is a key part of deep learning and the idea that symmetry can be learned directly from data has motivated several recent works on symmetry discovery.  The hope is that neural networks may be useful in e.g. scientific domains for discovering interpretable findings about different systems from data.\n- The method is an effective application of the theory of symmetries of differential equations.  The background is clearly presented and theorem 2.8 form [37] is nicely explained with an example.   \n- The method is able to explicitly find symmetries of learned models.  These symmetries are in the form of Lie algebra generators.  This is similar to the form of the symmetries found by (Dehmamy, 2021), however, an advantage of the current work is that no special architecture is necessary in order to learn the symmetry.  \n- It could be said the proposed method is not really for demonstrating that neural networks learn symmetry (see first weakness/question), but for learning invariances of the ground truth function from data.  At first, I wondered if these invariances could be estimated directly from data without training a neural network first.  However, it seems that training a neural network gives an effective way to estimate df/dx_i in equation 2 which would otherwise be difficult to do directly from data alone.  In this light, I think the proposed method is quite an effective way to discover invariances in data indirectly. \n- The finding (Line 244-245) that deeper networks learn symmetry better as the capacity is increased versus shallow networks is interesting and seems useful in network design. \n\n\n## Weakness / Questions\n- My main question about the method is that it's unclear what it means for a model to learn symmetry independently of learning the task function accurately.  Consider a regression problem.  If the model F trains well to approximate a ground truth function which is invariant with RMSE error at most \\epsilon, then it seems to me the symmetry variance (invariance error) is bounded by 2\\epsilon.  That is, the model has symmetry because it learned the ground truth and that has symmetry.  In this case, I would not really say the model did any particular invariance learning.  If it could be demonstrated that the model had even lower symmetry variance that expected given the accuracy or was capable of extrapolation based on the invariance, then I would grant the model had learned symmetry.  This question is somewhat examined in \"Symmetry vs. accuracy\" (Line 250), but here the conclusion seems to support my suspicion that any learned symmetry is really an emergent consequence of accurately fitting the ground truth function.      \n- I don't understand the second sentence of the abstract.  If a network first learns symmetry wouldn't it be starting to learn from no symmetry?  I presume the point of this sentence is perhaps to contrast with methods such as equivariant neural networks in which symmetry is encoded into the network as a prior, but I don't think it is very clear. \n- Theorem 3.1, Line 107-108. I looked at the reference [37] from which this theorem is quoted.  This statement appears to be a special case of the generality stated there, but with some missing assumptions.  The hypothesis here says n-dimensional manifold X, however, the equation given does not refer to local coordinates and so I believe assumes X=R^k and G \\subset GL_k(R) and acts by matrix multiplication on X.  If I understand correctly, this puts in the setting of [37, Example 1.28b] with a linear vector field.  That is, we are assuming symmetries are global linear symmetries of the space.  This is definitely still a useful level of generality to consider, but it does exclude some cases.  For example, F(x,y) = y - x^2.  I'd recommend clarifying the assumptions and deriving the special case from the general theorem in the appendix.  \n- Section 4.1 has some unclear writing.  The idea of image as function (Line 140-141) can be clearer.  Line 141:  A function should be defined either in terms of spaces, $X \\to Y$ or elements $x \\mapsto y$ or both.  This mixes both.  The notation in the equation between Line 146 and Line 147 is unclear.  Is the goal actually to describe a group action on the space of images?  In that case, I would define $(g \\cdot f)(x) = f(g^{-1} \\cdot x)$.  In particular, I would not use the same notation for the the group G and the action map $a \\colon G \\times X \\to X$.  Lastly, what is really going on here (it seems to me) is that the space of potential symmetries on the full space of images is too high-dimensional for the current method to be effective and so you have limited yourself to the 4-dimensional subgroup of spatial symmetries.  This is a strong prior over the potential invariances you are looking for. \n- Line 161:  It would be very helpful to include a formal definition of symmetry bias and symmetry variance. \n- Line 224-225: I'm not sure I agree with the conclusion about the smoothness of the spectrum of the polarization matrix or with the conclusion that the method can robustly find noisy symmetries.  In this case there is a very strong prior of the symmetry group being within a 4-dimensional subgroup of spatial symmetries. It's hard to evaluate smoothness over only 4 data points and the strength of this inductive bias could be the reason the model performs well here.    \n- \"Intermediate Symmetries\" (Line 276): In an equivariant neural network, only the last layer or last layers are usually used to project to an invariant representation.  Intermediate representations usually display equivariance instead of invariance with respect to some latent group representation type.  In that sense, it is perhaps not surprising that intermediate layers do not show invariance.  \n\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept