Original: LieGG: Studying Learned Lie Group Generators
Revised: LieGG: Studying Learned Lie Group Generators

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Symmetries built into the network have appeared to be very beneficial for a widerange of tasks as it saves  data to learn them.
Revised: 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.

Original: We depart from the position that whenthe symmetries are not built into a model a priori, it is advantageous for robustnetworks to learnthe symmetries directly from the data to fit the task function.
Revised: 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.

Original: In this paper, we present a method to extract symmetries learned by the neuralnetwork and to evaluate the degree to which the network is invariant to them.
Revised: 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.

Original: With our method, we are able to explicitly retrieve learned invariances in a formof the generators of corresponding Lie-groups without prior knowledge of thesymmetries in the data.
Revised: 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.

Original: We use the proposed method to study how symmetricalproperties depend on the network’s parameterization and configuration.
Revised: We use the proposed method to study how symmetrical properties depend on a neural network’s parameterization and configuration.

Original: We foundthat the ability of the network to learn symmetries generalizes over a wide rangeof architectures.
Revised: We found that the ability of a network to learn symmetries generalizes over a range of architectures.

Original: However, the quality ofthe learned symmetries depends on thedepth and  number of parameters.
Revised: However, the quality of learned symmetries depends on the depth and the number of parameters.

Original:  Introduction
Revised:  Introduction

Original: Convolutional Neural Networks (CNNs) are efficient, for one because they can convert translationsymmetry in the data into a built-in translation-equivariance property of the network without exhausting  data to learn the equivariance.
Revised: Convolutional Neural Networks (CNNs) are efficient, for one because they can convert the translation symmetry in the data into a built-in translation-equivariance property of the network without exhausting the data to learn the equivariance.

Original: Group-equivariant networks generalize this property to rotation[1, 2, 3, 4], scale [5, 6, 7], and other symmetries defined by matrix groups [8].
Revised: Group-equivariant networks generalize this property to rotation [6, 44, 41, 19], scale [37, 35, 4] and other symmetries defined by matrix groups [14].

Original: Equipping a neuralnetwork with prior known symmetries has proved to be data-efficient.
Revised: Equipping a neural network with prior known symmetries has proved to be data-efficient.

Original: Recent works [9, 10, 11] have demonstrated that hard coding the symmetry into the model does notalways lead to a better generalization.
Revised: Recent works [13, 20, 48] have demonstrated that hard coding the symmetry into a neural network does not always lead to a better generalization.

Original: Often, soft or learned equivariance is more advantageousmostly in terms of data efficiency and accuracy [12].
Revised: Often, soft or learned equivariance is more advantageous mostly in terms of data efficiency and accuracy [40].

Original: Moreover, architectures with no equivariancebuilt-in, such as transformers [13] and multi-layer perceptron mixers [14], achieve a remarkableperformance on a wide range of problems.
Revised: Moreover, architectures with no equivariance built-in, such as transformers [11] and multi-layer perceptron mixers [38], achieve a remarkable performance on a wide range of problems.

Original: Effectively, they learn their own geometrical priors withoutexplicit symmetry constraints [15].
Revised: Effectively, they learn their own geometrical priors without explicit symmetry constraints [8].

Original: This raises the following questions to study in this paper.
Revised: This raises the following questions to study in this paper.

Original: To whatdegree do neural networks learn symmetries?
Revised: To what degree do neural networks learn symmetries?

Original: How accurately do learned symmetries reflect the truesymmetries in the data?
Revised: How accurately do learned symmetries reflect the true symmetries in the data?

Original: Can we support the capability of  networks for learning symmetries?
Revised: Can we support the capability of neural networks for learning symmetries?

Original: Wepresent a method to study symmetries learned by  networks to address the questions.
Revised: We present a method to study symmetries learned by neural networks to address the questions.

Original: In the paper we depart from Lie group theory and develop a method that can retrieve symmetrieslearned from the data by any neural network.
Revised: 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 1).

Original: Our method  makes the assumption that the model isdifferentiable, commonly met in neural networks.
Revised: Our method only makes the assumption that a model is differentiable, commonly met in neural networks.

Original: While previous works on analyzing symmetries innetworks rely on empirical analysis of network representations [16] or on examination of a givenset of transformations [17, 6, 18], our method outputs a generator of the corresponding Lie-group toquantitatively evaluate how sensitive the network is in the direction of the learned symmetry(Figure1). We make the following contributions:
Revised: While previous works on analyzing symmetries in neural networks rely on empirical analysis of network representations [28] or on examination of a given set of transformations [15, 35, 25], our method outputs a generator of the 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:

Original: • From the perspective of Lie-groups, we propose a theory to study symmetrical properties ofa trained neural network.
Revised: • From the perspective of Lie-groups, we propose the theory to study symmetrical properties of neural networks.

Original: • We derive an efficient implementation  that allows improving the interpretability of the modelby revealing symmetries it learned.• With our method we conclude that models with more parameters and gradual fine-tuninglearn more precise symmetry groups with a higher degree of invariances to them.
Revised: • We derive an efficient implementation of the method that allows improving the interpretability 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 to them.

Original:  Related work 
Revised:  Related work Equivariant

Original: Equivariant and invariant networks The goal of equivariant and invariant networks is to buildthe symmetries into a  network architecture as an inductive bias.
Revised: and invariant networks The goal of equivariant and invariant networks is to build the symmetries into a neural network architecture as an inductive bias.

Original: Starting from the convolutionalnetworks [19] that contain  translation symmetry, the concept of  equivariance was generalized torotations [1, 2, 3, 4], permutations [20], scaling [5, 6, 21, 7, 22] and arbitrary matrix groups in [8].
Revised: Starting from the convolutional networks [24] that contain a translation symmetry, the concept of the equivariance was generalized to rotations [6, 44, 41, 19], permutations [46], scaling [37, 35, 36, 4, 43] and arbitrary matrix groups in [14].

Original: Various methods have been proposed for learning symmetries directly from the training data [11, 23]among which a popular approach is to learn transformations by estimating the infinitesimal generators[24, 25, 26, 27, 28].
Revised: Various methods have been proposed for learning symmetries directly from the training data [48, 3] among which a popular approach is to learn transformations by estimating the infinitesimal generators of symmetry groups [34, 10, 7, 33, 9].

Original: These papers focus on answering thequestion of how to build symmetries intothe models by modifying the architecture.
Revised: These papers focus on building the symmetries into the models by modifying the architecture.

Original: In our work, we focus on the reverse question, i.e. given anetwork with a fixed architecture, what symmetries does it learn from the data.
Revised: In our work, we focus on the reverse question, i.e. given a network with a fixed architecture, what symmetries does it learn from the data?

Original: Symmetries in neural networks Another line of work is focused on interpreting symmetriesin neural networks.
Revised: Symmetries in neural networks Another line of work is focused on interpreting symmetries in neural networks.

Original: In [16] Olah et al.
Revised: In [28] Olah et al.

Original: take inspiration from biological circuit motifs [29] andstudy the equivariant patterns learned by a neural network on the image classification task.
Revised: take inspiration from biological circuit motifs [2] and study equivariant patterns learned by an unconstrained neural network on the image classification task.

Original: Theauthors demonstrate that a network learns rotation, hue, and scale symmetries when trained on ImageNet.
Revised: The authors demonstrate that the network learns rotation, hue, and scale symmetries when trained on ImageNet.

Original: In [17] Goodfellow et al.
Revised: In [15] Goodfellow et al.

Original: propose a number of empirical tests to study invariancesto known transformations in the network, and demonstrate that auto-encoding architectures learnincreasingly invariant features indeep layers of the network.
Revised: propose a number of empirical tests to study invariances to known transformations in a network, and demonstrate that auto-encoding architectures learn increasingly invariant features in the deeper layers.

Original: Along the same lines, Lenc & Vedaldi[18] quantify invariances and equivariance in the layers of convolutional networks to a pre-definedset of transformations.
Revised: Along the same lines, Lenc & Vedaldi [25] quantify invariances and equivariance in the layers of convolutional networks to a pre-defined set of transformations.

Original: In this paper, we also propose a method to study the symmetrical propertiesof neural networks.
Revised: In this paper, we also propose a method to study the symmetrical properties of neural networks.

Original: In contrast to existing works, our method does not require knowing the setof transformations beforehand, and it also provides theoretical guarantees of invariances from theperspective of the Lie groups theory.
Revised: In contrast to existing works, our method does not require knowing the set of transformations beforehand; and it also provides theoretical guarantees of invariances from the perspective of the Lie groups theory.

Original: Network configuration: width, depth, number of parameters Neural networks of various widthsand depths have been studied through the lens of universal approximation theorem [30, 31], functionalexpressiveness [32] and by empirical analysis of learned representations [33].
Revised: Network configuration: width, depth, number of parameters Neural networks of various widths and depths have been studied through the lens of universal approximation theorem [26, 21], functional expressiveness [31] and by empirical analysis of learned representations [27].

Original: These works focuson either characterizingthe learning ability of the model or on interpreting the discrepancy betweenrepresentations that models with different architectures learn.
Revised: These works focus on either characterizing learning capabilities of neural networks or on interpreting the differences between representations that models with various architectures learn.

Original: They, however, do not analyze how thesymmetrical properties of a model depend on its width and depth.
Revised: They, however, do not analyze how the symmetrical properties of a model depend on its width and depth.

Original: Other works focus on analyzing the connection between a number of parameters and the generalization capability of networks [34, 35, 36, 37].
Revised: Other works focus on analyzing a connection between a number of parameters and the generalization capability of networks [1, 45, 39, 47].

Original: It is commonly argued that overparametrizedarchitectures achieve better generalization bounds, i.e. provide smaller discrepancy between train andtest performance and are thus preferred in many applications [37].
Revised: It is commonly argued that overparametrized architectures achieve better generalization bounds, i.e. provide smaller discrepancy between train and test performance and are thus preferred in many applications [47].

Original: However, it is unclear if modelswith more parameters and hence with a better generalization capability also learn better symmetries.
Revised: However, it is unclear if the models with more parameters and hence with a finer generalization capability also learn symmetries better.

Original: In our paper, we investigate this  for a family of feed-forward networks.
Revised: In our paper, we investigate this question for a family of feed-forward networks.

Original: Robust Learning Various methods have been proposed to train  robust models on small datasets.
Revised: Robust Learning Various methods have been proposed to train more robust neural networks.

Original: While the naive training may lead to overfitting on the training subset [38], well-organized pretraininghelps to mitigate this issue.
Revised: While a naive training may lead to overfitting on the training subset, a well-organized pre-training helps to mitigate this issue [12].

Original: Features extracted by a model pretrained on a bigger dataset oftendemonstrate more robust results [39].
Revised: Features extracted by a model pretrained on a bigger dataset often demonstrate more robust results [17].

Original: While there are many useful techniques for training effectivefeature extractors [40, 41, 42], the analysis of such methods from the invariance point of view wasnot done before.
Revised: While there are many effective techniques for training more robust feature extractors [5, 49, 16], the analysis of such methods from the invariance point of view has not been done before.

Original: We demonstrate that our method allows for better understanding and explainingwhat training regimes lead to more invariant, and thus more robust features.
Revised: We demonstrate that our method allows for better understanding and explaining what training regimes lead to more invariant, and thus more robust representations.

Original:  Background
Revised:  Background

Original: The focus of this paper is a symmetry group .
Revised: The focus of this paper is a symmetry group .

Original: Intuitively, a symmetry is a transformation of an objectthat leaves the object unchanged.
Revised: A symmetry is a transformation of an object that leaves the object unchanged.

Original: The set of all such transformations of an object with compositionas a binary operation forms a group.
Revised: The set of all such transformations of an object with composition as a binary operation forms a group.

Original: In this paper, the object of interest is a dataset, and we studywhat kind of transformations map one data sample to another and how neural networks preserveinformation about  symmetries.
Revised: In this paper, the object of interest is a dataset and its symmetries. We study what kind of transformations map one data sample to another and how neural networks learn information about the symmetries.

Original: The theory of Lie groups is a sweet spot of mathematics that helps to formalize symmetries andprovides practical methods for studying them.
Revised: The theory of Lie groups is a sweet spot of mathematics that helps to formalize symmetries and provides practical methods for studying them.

Original: Formally, a Lie group is a group that has the structureof a differential manifold.
Revised: Formally, a Lie group is a group that has the structure of a differential manifold.

Original: The tangent space in the identity element of a Lie group forms a vectorspace called Lie algebra.
Revised: The tangent space in the identity element of a Lie group forms a vector space called Lie algebra.

Original: A Lie algebra determines a Lie group up to an isomorphism for a simplyconnected Lie groups and, being a vector space, is a more convenient object to study than a Lie group.
Revised: A Lie algebra determines a Lie group up to an isomorphism for simply connected Lie groups and, being a vector space, is a more convenient object to study than a Lie group.

Original: Thus, in order to recover a simply connected symmetry group, it is sufficient to understand its Liealgebra.
Revised: Thus, in order to recover a simply connected symmetry group, it is sufficient to understand its Lie algebra.

Original: In this paper, we assume that a Lie group G is a matrix Lie group, i.e., a closed subgroup of thegeneral linear group  degree n , denoted as GL ( n ) and is defined as the set of n × n invertible matrices.
Revised: In this paper, we assume that the Lie group G is a matrix Lie group, i.e. is a closed subgroup of the general linear group of the degree n , denoted as GL ( n ) . It is defined as the set of n × n invertible matrices.

Original: The correspondence between the Lie group G and its Lie algebra g in this case is given by theexponential map:
Revised: The correspondence between the Lie group G and its Lie algebra g in this case is given by the exponential map:

Original: [Equation]
Revised: [Equation]

Original: where M ( n ) is the set of all n × n matrices.
Revised: where M ( n ) is the set of all n × n matrices.

Original: Each such element h from Equation 1 is called aninfinitesimal generator of the Lie group G .
Revised: Each such element h from equation 1 is called an infinitesimal generator of the Lie group G .

Original: We introduce LieGG , a method which extracts Lie G roup G enerators, i.e. finds the infinitesimalgenerators of the Lie algebra.
Revised: We introduce LieGG , a method to compute Lie G roup G enerators. LieGG extracts the infinitesimal generators of the Lie algebra for symmetries that a neural network learned from the data.

Original: To find the Lie algebra basis, we use the discriminator of the dataset,i.e. a function f : R n → R such that f ( x ) = 

Revised: To find the Lie algebra basis, we use the discriminator of the dataset, i.e. a function f : R n → R such that f ( x ) =

Original: if and only if x is an element of the dataset D data.
Revised: if and only if x is an element of the dataset D data.

Revised: The discriminator function is naturally modeled by a neural network fitting the data subject to a downstream task.

Original: Having a discriminator, we use the following criterion for a symmetry group of algebraic equations:
Revised: With this, we use the following criterion for a symmetry group of algebraic equations:

Original: Theorem 3.1 (Theorem 2.8 [43]) .
Revised: Theorem 3.1 (Theorem 2.8 [29]) .

Original: Let G be a connected Lie group of linear transformations actingon an n -th dimensional manifold X .
Revised: Let G be a connected Lie group of linear transformations acting on an n -th dimensional manifold X .

Original: Let F : X → R l , l ≤ n , define a system of algebraic equations:
Revised: Let F : X → R l , l ≤ n , define a system of algebraic equations:

Original: [Equation]
Revised: [Equation]

Original: and assume that the system is of maximal rank, meaning that the Jacobian matrix (cid:16) [Equation] ∂x k (cid:17) is of rank lfor every solution x of the system.
Revised: and assume that the system is of maximal rank, meaning that the Jacobian matrix (cid:16) ∂F ν ∂x k (cid:17) is of rank l for every solution x of the system.

Original: Then G is a symmetry group of the system if and only if 
Revised: Then G is a symmetry group of the system if and only if i = n,j = l (cid:88) i =1 ,j = ∂F ν

Original: [Equation]
Revised: [Equation]

Revised: = 0 , whenever F

Original: [Equation]
Revised: [Equation]

Original:  for every infinitesimal generator h of G , where h ij is an element of the matrix h in the i -th row andj -th column.
Revised: ( x ) = 0 , for every infinitesimal generator h of G , where h ij is an element of the matrix h in the i -th row and j -th column.

Original: Example Let us illustrate the practical significance of the theorem with an example.
Revised: Example We illustrate the practical significance of the theorem with an example.

Original: Suppose data D lie on a sphere in R n :
Revised: Suppose the dataset D lies on a sphere in R n :

Original: [Equation]
Revised: [Equation]

Original: The discriminator for this example is the function f = x 21 + · · · + x 2 n − 1 .
Revised: The discriminator for this example is the function f = x 21 + · · · + x 2 n − 1 .

Original: According to the theorem,in order to find  Lie algebra of the symmetry group, we can find a basis of the solution of the followinglinear equations, where a ij are sought variables:
Revised: 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 a ij are sought variables:

Original: [Equation]
Revised: [Equation]

Original: when x 21 + · · · + x 2 n = 1 .
Revised: when x 21 + · · · + x 2 n = 1 .

Original: We assert, that the solutions are  matrices A such thatA
Revised: We assert that the solutions are the family of matrices A such that

Original:  + A T = 0 .
Revised: A + A T = 0 .

Original: Indeed, for x : x i = 1 , x j = 0 , j ̸
Revised: Indeed, for x : x i = 1 , x j = 0 , j ̸

Original: = i , the condition means that a ii = 0 .
Revised: = i , the condition means that a ii = 0 .

Original: For x : x i = 1 √ 2 , x j =1 √ 2 , x k = 0 , k ̸
Revised: For x : x i = 1 √ 2 , x j = 1 √ 2 , x k = 0 , k ̸

Original: = i, j , it follows, that a ij + a ji = 0 .
Revised: = i, j , it follows that a ij + a ji = 0 .

Original: On the other hand if A = − A T , then(cid:80) i ̸
Revised: On the other hand, A = − A T implies (cid:80) i ̸

Original: = j (cid:0) x i · a ij · x j + x j · a ji · x i (cid:1)
Revised: = j (cid:0) x i · a ij · x j + x j · a ji · x i (cid:1)

Original: = (cid:80) i ̸
Revised: = (cid:80) i ̸

Original: =
Revised: =

Original: j (cid:0) x i · a ij · x j − x i · a ij · x j (cid:1) = 0 .
Revised: j (cid:0) x i · a ij · x j − x i · a ij · x j (cid:1) = 0 .

Original: Note that the Lie algebra of the matrices{ A
Revised: Note that the Lie algebra of the matrices { A : A

Original: [Equation] A + A T = 0 } corresponds to the Lie group of matrices [Equation] which is a rotation group.
Revised: + A T = 0 } corresponds to the Lie group of matrices { B : B · B T = E } which is the rotation group.

Original: Thus the symmetry of a sphere is a rotation group.
Revised: Thus, the symmetry of a sphere is the rotation group.

Original:  Method
Revised:  Method

Original: To compute a Lie algebra given a discriminator function f : R n → R we use Theorem 3.1 and solvethe system of linear equations, where each equation corresponds to one element in the dataset:
Revised: To compute a Lie algebra given a discriminator function f : R n → R we use Theorem 3.1 and solve the system of linear equations, where each equation corresponds to one element in the dataset:

Original: [Equation]
Revised: [Equation]

Original: each element from the dataset.
Revised: the dataset

Original: [Equation]
Revised: [Equation]

Original: This is a system of linear equations with n 2 variables, and  number of equations  equal to the numberof points in the data.
Revised: This is a system of linear equations with n 2 variables, and the number of equations is equal to the number of points in the data.

Original: We denote the matrix defined by equations 2 as E and call it the networkpolarization matrix .
Revised: We denote the matrix defined by equations 2 as E and call it the network polarization matrix .

Original: The problem of finding a basis of solutions of linear equations is a standardproblem in  numerical analysis, that can be solved using singular value decomposition (SVD).
Revised: The problem of finding a basis of solutions to linear equations is a standard problem in the numerical analysis, that can be solved using the Singular Value Decomposition (SVD).

Original: Weremind that we can write  E using  SVD in the following way: [Equation] where U, V - are orthogonal matrixes and Σ is a diagonal matrix with decreasing elements on thediagonal.
Revised: Recall that we can write matrix E using the SVD in the following way E = U Σ V T , where U, V - are orthogonal matrices and Σ is a diagonal matrix with decreasing singular values. Thus, the columns in

Original: Thus, the columns in V corresponding to almost 0 singular values  form a Lie algebra basis.
Revised: V corresponding to nearly 0 singular values encode the nullspace of the system of equations in 2 and hence form a Lie algebra basis.

Original: Thereby, in practice, the calculation of LieGG consists of 3 steps: 1.training a neural network;2.calculation of the polarization matrix E ; and 3.calculation of singular vectors corresponding toalmost zero singular values of the  matrix E .
Revised: 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 .

Original: 4.1 (revised) Computation of Lie algebra of the group acting on R2 .
Revised: 4.1 Computing a Lie algebra of a group acting on R

Original: Symmetries play an important role in computer vision problems, and we will describe LieGG for this case in more detail.
Revised: Symmetries play an important role in computer vision problems, and we will describe LieGG for this case in more details.

Original: In various computer vision problems, it is assumed that the symmetry group G changes images from the dataset by acting on R 2 .
Revised: In various computer vision problems, it is assumed that the symmetry group G changes images from the dataset by acting on R 2 .

Original: For example, convolutional networks [19] and recently proposed equivariant networks for rotations [3] assume that the group acts as a translation or rotation of the image.
Revised: For example, convolutional networks [24] and recently proposed equivariant networks for rotations [44, 41] assume that the group acts as a translation or rotation of an image.

Original: These neural networks perceive each image as a function f with non-zero values on some compact subset Ω ⊂ R 2 , where the elements of the subset correspond to spatial coordinates.
Revised: These neural networks perceive each image as a function f with non-zero values on some compact subset Ω ⊂ R 2 , where the elements of the subset correspond to spatial coordinates.

Original: Thus, the space of images X is the space of all such functions:
Revised: Thus, the space of images I is the space of all functions that map spatial coordinated to pixel values.

Original: [Equation]
Revised: In other words: I := { f, f : Ω → R } .

Original: A vision model F : X → R is a function on such a space of functions.
Revised: A vision model F : I → R is a function on such a space of images.

Original: We view an image is afunction on a pixel grid ( x 1 , · · · , x n ) of the domain Ω , i.e. the image can be identified with thepoint in R n , which encode the pixel values.
Revised: We view an image is a function on a pixel grid ( x 1 , · · · , x n ) of the domain Ω , i.e. the image can be identified with the point in R n , which encodes the pixel values.

Original: In this case the model takesas input n real values  F ( f ( x 1 ) , · · · , f ( x n )) .
Revised: In this case the model takes n real values as an input: F ( f ( x 1 ) , · · · , f ( x n )) .

Original: It is important to note that Theorem 3.1 assumes that the group G acts continuously on the datamanifold X .
Revised: It is important to note that Theorem 3.1 assumes that the group G acts continuously on the data manifold.

Original: However, this assumption is violated for  image data as an image is a non-continuousfunction of coordinates due to the discrete pixel grid.
Revised: However, this assumption is violated for the image data as an image is a non-continuous function of coordinates due to the discrete pixel grid.

Original: We overcome the discontinuity issue bypreprocessing images with Gaussian smoothing.
Revised: We overcome the discontinuity issue by preprocessing images with Gaussian smoothing.

Original: To use LieGG for  image data, we explicitly write the condition that the matrix h is an infinitesimalgenerator for the image data X on the grid ( x 1 , · · · , x n ) , where x p [ j ] corresponds to the projectionof x p onto the j-th coordinate axis.
Revised: To use LieGG for the image data, we explicitly write the condition that the matrix h is an infinitesimal generator for the image data I on the grid ( x 1 , · · · , x n ) .

Original: The condition reads as follows:
Revised: The condition reads as follows:

Original: F

Original: [Equation]
Revised: [Equation]

Original: p = n,k =2 ,j =2 (cid:88) p =1 ,k =1 ,j = ∂F ∂f ( x p ) · ∂f ( x p ) ∂ x p [ k ] · h k,j · x p [ j ] = 0 (3) where t ∈ R .
Revised: where t ∈ R and x jp corresponds to the projection of x p onto the j-th coordinate axis.

Original: The resulting condition is the equation in 4 variablesh k,j , k
Revised: The resulting condition is the equation in 4 variables

Original:  = 1 , 2; j = 1 , 2 .
Revised: h kj , k = 1 , 2; j = 1 , 2 .

Original: Thus, in this case, the polarization matrix has 4 columns.
Revised: Hence, the polarization matrix hascolumns.

Original: Thus, in equation 3 a space of potentialimage symmetries is reduced to  4-dimensional spatial subgroup.
Revised: Thus, in equation 3 a space of potential image symmetries is reduced to a 4-dimensional spatial subgroup.

Original: This is to provide a prior forthe LieGG to retrieve meaningful symmetries from for images as the full-scale space of the potentialsymmetries is overwhelming for  image data.
Revised: This is to provide a prior for LieGG to retrieve meaningful symmetries from for images as the full-scale space of potential symmetries is overwhelming for the image data.

Original: This is a strong yet proven to useful prior for visionmodels [1, 44, 2, 3, 4].
Revised: This is a strong yet proven to useful prior for vision models [6, 18, 44, 41, 19].

Original: In practice, the calculation of LieGG for vision models consists of the following steps: (i) preprocessing images by doing Gaussian smoothing, (ii) training a neural network, (iii) computing polarizationmatrix using equation 3.
Revised: In practice, the calculation of LieGG for vision models consists of the following steps: (i) preprocessing 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.

Original: 4.2 Symmetry bias and symmetry variance
Revised: 4.2 Symmetry bias and symmetry variance

Original: The scope of LieGG is not limited to  the study of symmetry, but also to the evaluation of thedegreeof neural network invariance.
Revised: The scope of LieGG is not limited to retrieving the generator of a learned symmetry group. With the polarization matrix E , we can also evaluate the degree of the neural network’s invariance to a learned symmetry.

Original: If we know the true symmetries, we can apply our method to analyzehow close the extracted symmetries are to the true ones; and the degree of network invariance to thesesymmetries.
Revised: Furthermore, if a true symmetry in the data is known, we can apply our method to analyze how close is a learned symmetry to the true one.

Original: The former  we alias symmetry bias , and the later - symmetry variance .
Revised: The former criterion we alias symmetry variance , and the later - symmetry bias .

Original: To understandthis we show how we can estimate the degree of  network invariance to the element g of a Lie group:
Revised: Firstly, we show how to estimate the degree of the network invariance to the element g of a Lie group:

Original: [Equation]
Revised: [Equation]

Original: Let h is an element from Lie algebra, that corresponds to g , i.
Revised: Let h be an element from Lie algebra that corresponds to g , i.e.

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: We denote the matrix with the i -th row equal to ∂F ( x i ) ∂x k · x ij as E .
Revised: The matrix with the i -th row equal to ∂F ( x i ) ∂x ki · x ji equates to the polarization matrix E .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where  h is a vector with coordinates equal to h kj .
Revised: where ¯ h is the vectorized representation of the element of the Lie algebra.

Original: To calculate the value of E · h we use  SVD ofmatrix E .
Revised: To calculate the value of E ¯ h we use the SVD of the matrix E .

Original: Lets E = U Σ V ∗ , where U, V - are semi-unitary matrixes and Σ is a diagonal matrix, andh = (cid:80) l h l v i .
Revised: Let E = U Σ V T , where U, V - are semi-unitary matrices and Σ is a diagonal matrix.

Original: We can estimate ˜ R ( g ) in the following way:
Revised: With this, we can estimate ˜ R ( g ) in the following way:

Original: [Equation]
Revised: [Equation]

Original: where σ l is the l -th singular value.
Revised: where σ l is the l -th singular value of the network polarization matrix E .

Original: In particular, the equation 4 shows that singular values  reflect the degree ofnetwork invariancetotransformations corresponding to  singular vectors.
Revised: In particular, the equation 4 reveals that singular values of the network polarization matrix reflect the degree of invariance to transformations encoded by the corresponding singular vectors.

Original: With this, we use the smallest singular value ofthe network polarization matrix to define the symmetry variance as V ( E ) =
Revised: With this, we use the smallest singular value of the network polarization matrix to define the symmetry variance as V ( E ) =

Original: |E| − 1 σ min ( E ) 2 , where|E| denotes number of rows in the network polarization matrix.
Revised: σ min ( E ) 2 / | D | .

Original: Moreover, when the symmetries areknown, the difference between the singular vectors and the generator of the known symmetry groupequates to the symmetry bias , i.e. B i ( E , G ) =
Revised: Moreover, when the symmetries are known, the difference between the singular vectors and the generator of the known symmetry group equates to the symmetry bias , i.e. B i ( E , G ) =

Original: ||
Revised: ||

Original: ˜ v i ( E ) −˜ G|| F , where ˜ v i ( E ) returns the i-th singularvector reshaped to a matrix, and˜ G is a ground truth symmetry group generator, which is the closestto ˜ v i ( E ) , e.g. the closest skew-symmetric matrix for O ( n ) .  Experiments 5.1 Implementation details
Revised: ¯ v i ( E ) − G|| F , where ¯ v i ( E ) returns the i-th singular vector reshaped to a matrix, and G is a ground truth symmetry group generator closest to the ¯ v i ( E ) , e.g. the closest skew-symmetric matrix for the rotation group.

Original: We use a multi-layer perceptron with linear layers followed by Swish activation [45].

Original: For theregression experiment, we use the MLP with 3 layers and set the hidden dimension to 32 .

Original: For theclassification experiment we use the MLP with a classification head attached, and train variousconfigurations varying a number of parameters, depth, and width.

Original: We train all of the models with

Original: Adam optimizer [46] setting the learning rate set to 10 − 3 .

Original: The results for all of the experiments areaveraged over 8 common fixed random seeds.

Original:  5.2 Retrieving symmetries from a synthetic regression
Revised:  Experiments 5.1 Retrieving symmetries from the synthetic regression

Original: Firstly, we perform the experiment on a synthetic problem to test the ability of our method toaccurately retrieve the symmetry learned by the network.
Revised: Firstly, we conduct the experiment on the synthetic problem to test the ability of our method to accurately retrieve a symmetry learned by a network.

Original: To do so, we adopt the invariant regressiontask from [8] with clean O (5) symmetry built-in.
Revised: To do so, we adopt the invariant regression task from [14] with clean O (5) symmetry built-in.

Original: The regression function f : R 2 × 5 → R is given by:
Revised: The regression function f : R 2 × 5 → R is given by:

Original: [Equation]
Revised: [Equation]

Original: With this, we construct the training dataset to fit the regression function with the multi-layer perceptron.
Revised: With this, we construct the training dataset to fit the regression function with the multi-layer perceptron.

Original: We note that the MLP is an unconstrained feed-forward neural network with no a priori knowledge ofsymmetries in the data.
Revised: Here the MLP is the unconstrained feed-forward neural network with no prior knowledge of symmetries in the data.

Original: Once the model is converged, we apply our method to analyze how close arethe retrieved symmetries tothe clean O (5) group (symmetry bias), and how invariant is the networkto these symmetries (symmetry variance).
Revised: Once the model is converged, we apply our method to analyze how close are retrieved symmetries to clean O (5) group (symmetry bias), and how invariant is the network to these symmetries (symmetry variance).

Original: We also study how the ability of the network to learnsymmetries depends on a number of available training samples.
Revised: We also study how the ability of the network to learn symmetries depends on the number of available training samples.

Original: We report the results in Figure 2.
Revised: We report the results in Figure 2.

Original: As can be seen from Figure 2a, the spectrum of the network polarization matrix is zeroing, indicatingthat the model becomes invariant to the transformations encoded by the corresponding singularvectors, and these singular vectors are the generators of the rotation group as indicated by zero O (5)symmetry bias  (Figure 2b).
Revised: As can be seen from Figure 2a, the spectrum of the network polarization matrix is zeroing, indicating that the model becomes invariant to 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 to O (5) group (Figure 2b).

Original: We also observed that the networkcan accurately learn the symmetrygroup generators even on a small training set size.
Revised: We also observed that the network accurately learns the symmetry group generators even with a small training set size.

Original: However, a larger training set size results in thenetwork having smaller symmetry variance, i.e. being more invariant.
Revised: However, a larger training set size leads to a smaller symmetry variance, i.e. a higher degree of invariance.

Original: For  additional validation, weused the LieGG to extract symmetries from the randomly initialized O (5) invariant EMLP [8].
Revised: For an additional validation, we used LieGG to extract symmetries from the randomly initialized O (5) invariant EMLP [14] model.

Original: Forthe EMLP, the LieGG recorded zero symmetry variance and zero symmetry bias with respect tothe O (5) group without any model pre-training.
Revised: For this case, LieGG recorded zero symmetry variance and zero symmetry bias with respect to O (5) group without any model pre-training.

Original: This experiment demonstrates that on a synthetic  task, our method can accurately retrieve thesymmetries learned or built-in.
Revised: The experiment demonstrates that, on the synthetic regression task, our method can accurately retrieve symmetries learned or built-in.

Original: Whenthe symmetries are learned, the larger training dataset facilitateslearning invariances for the model.
Revised: When symmetries are learned, the larger training dataset facilitates learning invariances for the model.

Original: 5.3 Retrieving symmetries from  rotation MNIST
Revised: 5.2 Retrieving symmetries from the rotation MNIST

Original: In this experiment, we test the ability of our method to retrieve learned symmetries on a morechallenging rotation MNIST dataset [47].
Revised: In this experiment, we test the ability of our method to retrieve learned symmetries on the rotation MNIST dataset [23].

Original: We train the feed-forward network to output the class of the digit invariant to 2D in-plane rotations.
Revised: We train the feed-forward network to output the class of a digit invariant to 2D in-plane rotations.

Original: The symmetry group that the model should learn is thus SO (2) .
Revised: The symmetry group that the model should learn is thus SO (2) .

Original: We note that different from the synthetic dataset, the symmetries in the rotation MNIST aremuch noisier since the rotations are coarsened by the projection onto a pixel grid.
Revised: 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.

Original: In addition, thenetwork has a classification error, hence it serves as only an approximate data discriminator function.
Revised: In addition, the network has a classification error, hence it serves as only an approximate data discriminator function.

Original: In this scenario, we test how well our method can retrievethe noisy symmetries.
Revised: In this scenario, we test how well our method can retrieve noisy SO (2) symmetry.

Original: We experiment with two architectures: a shallow 2-layer perceptron and a deeper 6-layer networkwith 4 times number of parameters.
Revised: We experiment with two architectures: the shallow 2-layer perceptron and the deeper 6-layer network with 4 times number of parameters.

Original: Once, the networks’ performance plateaus on the validationsplit, we terminate the training, and apply our method to retrieve learned symmetries.
Revised: Once the network’s performance plateaus on the validation split, we terminate the training, and apply our method to retrieve learned symmetries.

Original: We plot thespectrum of the network polarization matrix, SO (2) symmetry bias and visualize learned symmetriesfor shallow and deep models in Figure 3.
Revised: 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.

Original: From Figure 3a we observe that for both shallow and deep models, our method can robustly retrievethe rotation symmetry as indicates by  zeroing symmetry bias with respect to SO (2) group, given the4-dimensional spatial subgroup prior.
Revised: From Figure 3a we observe that for both shallow and deep models, our method retrieves the rotation symmetry as indicated by the zeroing symmetry bias with respect to SO (2) group.

Original: However, we noticed that the rotation learned by the shallowmodel isin general noisier than the symmetry learned by the deeper networks as indicated by a highersymmetry bias.
Revised: Also, we noticed that the symmetry learned by the shallow model is noisier than the symmetry learned by the deeper network as indicated by the higher symmetry bias.

Original: We visualize it in Figure 3b by sampling learned Lie algebras and applying them totransform an input image.
Revised: We visualize this in Figure 3b by sampling learned Lie algebras and applying them to transform an input image.

Original: We observe that the symmetry learned by the shallow model diverges froma clean rotation and also contains scaling and shearing.
Revised: We see that the symmetry learned by the shallow model diverges from a clean rotation and also contains scaling and shearing.

Original: We also see, that on  rotation-MNIST, thespectrum of the network polarization matrix is zeroing not as fast as in the synthetic case.
Revised: We also see, that on the rotation-MNIST, the spectrum of the network polarization matrix is zeroing not as fast as in the synthetic case.

Original: It indicatesthat the models, although do learn the rotation symmetry, do not become fully invariant to it.
Revised: It indicates that the models, although do learn the rotation symmetry, do not become fully invariant to it.

Original: This experiment demonstrates that our method canrobustly retrievenoisy symmetries from the dataeven when an imprecise discriminator function is used.
Revised: This experiment demonstrates that our method can retrieve symmetries from the data even when an imprecise discriminator function is used.

Original: 5.4 Symmetries in networks with different configurations
Revised: 5.3 Symmetries in networks with different configurations

Original: Observing the differences in symmetrical properties for different network configurations in Experiment 5.3, we further conduct the study on how  symmetry variance and  symmetry bias depend onwidth, depth, and a number of parameters in the network.
Revised: Observing the differences in symmetrical properties for different network configurations in Experiment 5.2, we further conduct the study on how the symmetry variance and the symmetry bias depend on width, depth, and a number of parameters in a neural network.

Original: In this experiment, we are interested toknow if for some architecture it is easier to learn symmetries from the data.
Revised: In this experiment, we are interested to know if for some network configurations it is easier to learn symmetries from the data.

Original: To obtain different configurations, we sample a number of parameters from a range of [40000 , 200000]with a step  of 20000 , and vary the depth of the network from 1 to 5 hidden layers.
Revised: To obtain different configurations, we sample a number of parameters from the range of [40000 , 200000] with the step size of 20000 , and vary the depth of a network from 1 to 5 hidden layers.

Original: We train thenetworks until convergence on the validation split and apply our method to study how well theylearn the symmetries.
Revised: 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.

Original: We also record the accuracy of the models to correlate it with their symmetryvariance and  symmetry bias.
Revised: We also track the accuracy of the models to correlate it with the symmetry variance and the symmetry bias.

Original: We report results  in Figure 4.
Revised: The results are reported in Figure 4.

Original: From Figure 4a we can see that we make the following conclusions on how the configurations of thenetwork influence its symmetrical properties:
Revised: We summarize the analysis of how the configurations of the network influence its symmetrical properties:

Original: Number of parameters: we observe that the networks with more parameters  learn to become moreinvariant than their shallow counterparts as indicated by consistently smaller symmetry variance.
Revised: Number of parameters: we observe that the networks with more parameters tend to learn to the higher degree of invariance than their shallow counterparts as indicated by consistently smaller symmetry variance.

Original: Also,we see that the networks with fewernumber of parametersgenerally learn less precise symmetries asshown by  higher symmetry bias.
Revised: Also, we see that the networks with fewer parameters learn less precise symmetries as indicated by the higher symmetry bias.

Original: Depth / width: our results indicate thatthe sufficiently parameterized deep networks are better ableto learn invariances than the wide ones with the same amount of parameters, i.e. deeper the networkis, more invariantit becomes in the direction ofthe learned symmetries.
Revised: Depth / width: our results indicate that sufficiently parameterized deep networks are more capable to learn invariances than the wide ones with the same number of parameters. In other words, deeper networks become more invariant in the direction of learned symmetries.

Original: At the same time, this doesnot hold for the quality of  learned symmetry groups as indicated by  fluctuating symmetry bias, whendepth > 3 .
Revised: At the same time, the same only partially holds for the quality of a learned symmetry group as indicated by the fluctuating symmetry bias, when the depth > 3 .

Original: We see that the network with  balanced width and depth achieve smaller symmetry biasthan the deeper ones.
Revised: Overall, the network with a balanced width and depth achieves the smallest symmetry bias.

Original: Symmetry vs. accuracy: we also test how the symmetry variance and  bias correlates with theperformance of the model.
Revised: Symmetry vs. accuracy: we also test how the symmetry variance and the symmetry bias correlate with the performance of the models.

Original: Aswe can see from Figure 4b, the symmetry variance and bias havemoderate correlation coefficients ρ var = − 0 .
Revised: As can be seen from Figure 4b, the symmetry variance and bias have moderate negative correlation coefficients ρ var = − 0 .

Original: 49 ( p-value = 10 − 35 ) and ρ bias = − 0 .
Revised: 49 ( p-value < 1 e − 4 ) and ρ bias = − 0 .

Original: 31 ( p-value =10 − 14 )  respectively.
Revised: ( p-value < 1 e − 4 ) with the classification accuracy respectively.

Original: This trend indicates that more accurate modelsgenerally also tend to have lowersymmetry variance and  bias.
Revised: The trend indicates that more accurate models tend to have lower symmetry variance and lower symmetry bias.

Original: However such correlation is not strict, and it is possible to have anetwork, whichdoes not learnthe most precise symmetries from the training data, but still performwell on the test.
Revised: However, the correlation is not strict. That is, it is possible to have neural networks, which learn symmetries differently, but perform equally well during the test.

Original: 5.5 Tuningthe layer-wise symmetries in deep networks
Revised: 5.4 Tuning layer-wise symmetries in deep networks

Original: In the previous experiment we analyzed how the symmetrical properties of the networks dependon the configuration of the model.
Revised: In the previous experiment, we analyzed how the symmetrical properties of the networks depend on the configuration of the model.

Original: Now, we ask if by modifying  training regime, we can facilitatelearning the symmetries for the network.
Revised: Now, we ask if by modifying a training regime, we can facilitate learning symmetries for the network.

Original: In this experiment we test two known approaches to improvethe training: (i) pre-training with fine-tuning, and (ii) layer-wise training approach [40].
Revised: In this experiment, we evaluate two known approaches to robust training: (i) pre-training with fine-tuning, and (ii) layer-wise training [5].

Original: To simulate the pre-training  a part of the network weights, we take randomly initialized 7 layersperceptron (mother network), and extract  sub-network that consist of the first 3 layers.
Revised: To simulate the pre-training of a 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.

Original: We then attachthe classification head to this sub-network and train it independently from the mother network onrotation-MNIST dataset.
Revised: We then attach the classification head to the sub-network and train it independently from the mother network on the rotation-MNIST dataset.

Original: After that, we re-inject the learned weights into the mother network andretrain the model.
Revised: After that, we re-inject the learned weights into the mother network and retrain the model.

Original: For  fine-tuning, we train all of the weights jointly.
Revised: For the fine-tuning, we train all of the weights jointly.

Original: For  layer-wise training, wefreeze the pre-trained part and only train the remaining layers.
Revised: For the layer-wise training, we freeze the pre-trained part and only train the remaining layers.

Original: We then apply our method to evaluatethe symmetrical properties of the models.
Revised: We then apply our method to evaluate the symmetrical properties of the models.

Original: We plot the symmetry variance and  symmetry bias aftereach layer of the network in Figure 5.
Revised: We plot the symmetry variance and the symmetry bias after each layer of the network in Figure 5.

Original: Final and pen-ultimate layer symmetries: from Figure 5 we observe that for  different trainingregimes, the network learns approximately the same symmetries at the same degree of invariance in afinal layer.
Revised: 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.

Original: Surprisingly, we also noticed that in a pen-ultimate layer, the symmetries for all of thetraining regimes come very close as measured by  symmetry variance and bias.
Revised: Surprisingly, we also noticed that in the pen-ultimate layer, the symmetries for all of the training regimes come very close as measured by the symmetry variance and bias.

Original: Since the last layersare directly responsible for providing outputs for the task 1 , we hypothesize that there exist  preferred optimal symmetry level for the whole network that depends on the architecture and  data, but not fromthe training regime of the model .
Revised: Since the last layers are directly responsible for providing outputs for the downstream task, we hypothesize that there exist a preferred optimal symmetry level for a neural network that depends on an architecture and the data, but not from a training regime of a model .

Original: Intermediate symmetries: on the other hand, we observed thatthe symmetries in the intermediateslayers of the network do depend on the training regime .
Revised: Intermediate symmetries: on the other hand, we observed that symmetries in the intermediate layers of the network do depend on the training regime .

Original: Firstly, the pre-trained sub-network learnsmore accurate symmetry with a higher level of invariance compared to when it is trained as a part ofthe deeper model.
Revised: Firstly, the pre-trained sub-network learns more accurate symmetry with the higher level of invariance compared to when it is trained as a part of the deeper model.

Original: We attribute it to the fact that symmetries are defined on the level of the labels,hence the invariances learned by the model in the final layer should be kept at a minimum.
Revised: 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 at a minimum [42].

Original: Secondly,after the fine-tuning or  layer-wise training, symmetries in the intermediate layers of the network areboth more precise and utilized at a higher degree of invariance as indicated by  smaller symmetry biasand variance respectively.
Revised: Secondly, after the fine-tuning or the layer-wise training, symmetries in the intermediate layers of the network are both more precise and utilized at the higher degree of invariance as indicated by the smaller symmetry bias and variance respectively.

Original: It indicates that we can facilitate learning intermediate invariances in thenetwork by using  pre-training followed by  fine-tuning or  layer-wise learning approach.
Revised: 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.

Original: We assumethis insight can be useful for  transfer learning when it is important to transfer as much inductive biasas possible intothe other models or to other tasks.
Revised: This can be useful for a transfer learning when it is important to transfer as much inductive bias as possible into other models or to other tasks.

Original:  Conclusion
Revised:  Conclusion

Original: Summary We presentthe LieGG - an approach to retrievethe symmetries learned by a neuralnetwork.
Revised: Summary We present LieGG - the approach to retrieve symmetries learned by neural networks.

Original: Conveniently, our method extractsthe symmetries directly in a form of corresponding Lie group generators.
Revised: Conveniently, our method extracts symmetries directly in the form of corresponding Lie group generators.

Original: This allows to explicitly obtainthe symmetries learned by the model withoutspecifying any set of transformations of interest beforehand.
Revised: This allows to explicitly obtain symmetries learned by a neural network without specifying any set of transformations of interest beforehand.

Original: The LieGG also evaluates how themodel is invariant in thedirection of a learned symmetry by computing the symmetry variance.
Revised: 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.

Original: Wemake use of our approach by studying how the quality of the symmetry learned by the networkdepends on its configuration and a training regime.
Revised: 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.

Original: With this, by improving an interpretability of thesymmetry learning, wecan also identify good practices that improve the symmetries in a model.
Revised: By improving the interpretability of symmetry learning, we identify configurations and training regimes that facilitate training invariances in neural networks.

Original: Limitations and Future Work
Revised: Limitations and Future Work

Original: The proposed method can facilitate the interpretability of neuralnetworks and help to design efficient ways of incorporatingthe symmetries into the model.
Revised: The proposed method can facilitate the interpretability of neural networks and help to design efficient ways of incorporating symmetries into a model.

Original: However,some limitations remain.
Revised: However, some limitations remain.

Original: In particular,the LieGG is currently limited tothe instances of GL connectedgroup, which hinders the analysis of complex symmetries, e.g. where the transformation can not bedescribed as a group action.
Revised: In particular, LieGG is currently limited to instances of GL ( n ) connected group, which hinders the analysis of complex invariances, e g. where a transformation can not be described as a group action.

Original: It is an intriguing direction for future research to extend the mechanismof retrievingthe symmetries to more complex transformations.
Revised: It is an intriguing direction for future research to extend a mechanism of retrieving symmetries to more complex transformations.

Original: Nonetheless,the LieGG provides animportant building block on the way to increasing the interpretability of the symmetries learned bythe neural networks.
Revised: Nonetheless, LieGG provides an important building block on the way to increasing the interpretability of symmetry learning in neural networks.

Original: Also, in our analysis, we focus on a specific family of models, i.e. feed-forwardmulti-layer perceptions trained on the  well-controlled datasets.
Revised: 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.

Original: We assume thatthe further large-scaleanalysis ofthe LieGG in complex scenarios and with more advanced models(e.g. graph networks,and self-attention transformers) can provide a more general understanding ofthe symmetry learningin deep networks.
Revised: 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.

Original:  Checklist
Revised:  Acknowledgments and Funding Disclosure

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: We would like to thank Evgenii Egorov and Volker Fischer for their valuable comments and discussions. We thank the reviewers for their insightful feedback and helpful suggestions that improved our paper.

Original: [Yes]
Revised: 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.

Original: (b) Did you describe the limitations of your work? 
Revised: A Supplementary material A.1 Derivation of Theorem 3.

Original: [Yes] (c) Did you discuss any potential negative societal impacts of your work?
Revised: Let G be a connected Lie group of transformations acting on the n -th dimensional manifold X .

Original: [N/A]
Revised: Let

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]
Revised: F : X → R l , l ≤ n . With this, we define a system of algebraic equations: [Equation]

Original: If you are including theoretical results...
Revised: and assume that the system is of maximal rank, meaning that the Jacobian matrix (cid:16) ∂F ν ∂x k (cid:17) is of rank l for every solution x of the system. Then G is a symmetry group of the system if and only if: h [ F ν ( x )] = 0 , whenever F ν ( x ) = 0 , for every infinitesimal generator h of G .

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: In our special case, the infinitesimal generator h = { h ij } generates the one parameter group of transformations equal to exp( h · t ) . That means that in the theorem, h is an element of the symmetry group if and only if dF ν (exp [Equation]

Revised: h

Original: [Yes]{+[Equation] x [Equation] | t =
+}
Revised: {+[Equation] x [Equation] | t =
+}

Original: (b) Did you include complete proofs of all theoretical results?
Revised: = 0 , whenever F ν ( x ) = 0 . But [Equation] A.2 LieGG sample complexity

Original: [Yes]
Revised: LieGG computation requires finding a nullspace of the network polarization matrix. The dimensionality of the network polarization matrix depends on the number of samples in a dataset. Since the dimensionality of a full-scale dataset may be prohibitively large, we conduct the sample complexity study to investigate if the usage of all samples in a dataset is necessary to effectively retrieve the symmetries learned by a neural network. We conduct the empirical sample complexity study for O (5) synthetic regression and rotation-MNIST classification tasks.

Revised: For the synthetic regression, we take the trained model from A.3.1 and compute the symmetry variance and bias using only a fraction of the training samples.

Revised: We then compare the results with the symmetry variance and bias computed from all available training data.

Revised: The resulting empirical sample complexity is presented in Figure 6.

Revised: For the rotation-MNIST, we follow the same procedure and study the sample complexity of LieGG applied to retrieve symmetries from the trained 6-layer perceptron from A.3.2.

Revised: The resulting empirical sample complexity for the rotation-MNIST is presented in Figure 7.

Revised: Note that in both cases, the model is trained using all available training data.

Revised: The sample complexity study is only to investigate how many samples are needed to effectively extract the symmetries from the trained model.

Revised: For the synthetic regression task, it suffices to use only 0 .

Revised: 25% of the training samples to extract the symmetry as precisely as when utilizing all available training data.

Revised: For the more challenging rotation-MNIST, we have to use at least 5% of the training samples to obtain a precise estimation of the symmetry bias.

Revised: The results suggest that we can effectively extract the learned symmetry using only a fraction of the data to create the network polarization matrix.

Original: If you ran experiments...
Revised: Here we provide additional experiments and details on the experiments in the main paper. All of the experiments are performed with Pytorch 1.7.0 [30] on Nvidia Titan X GPU with the CUDA version 10.1 . The results and error bars are reported over 8 fixed random seeds [1-8] .

Revised: A.3.1 Retrieving symmetries from the synthetic regression

Revised: To fit the regression function we employ the multi-layer perceptron that consists of linear layers followed by Swish [32] activations with the following dimensions: (10 × 32) → 3 ∗ (32 × 32) → (32 × 1) .

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: The network takes input coordinates and outputs the predicted function value. We use a mean squared error loss to supervise the model. We utilize Adam optimizer [22] with a learning rate of 10 − 3 and train the network for a total of min (900000 /N, 1000) epochs, where N is the size of the training dataset. [14] reports this training schedule is enough for convergence.

Revised: To compute the mean symmetry variance per a training set size, we average the last 10 singular values of the network polarization matrix.

Revised: These singular values correspond to the O (5) symmetry as indicated by the corresponding symmetry biases.

Revised: A.3.2 Retrieving symmetries from the rotation-MNIST

Revised: We employ two models: shallow (2-layer) and deep (6-layer) perceptrons with 40000 andparameters respectively.

Revised: The hidden dimensions for the shallow and deep networks are 47 andrespectively.

Revised: Both models are trained for 300 epochs with Adam optimizer with the learning rate set to 10 − 3 .

Revised: We use a cross-entropy loss to supervise the model.

Revised: Symmetries are extracted from the models with the highest validation accuracy over 300 training epochs.

Revised: Additionally, to account for a possible difference in the magnitudes of the network outputs, we normalize the network output logits to be unit L2 norm.

Revised: A.3.3 Retrieving symmetries from the rotation-augmented CIFAR

Revised: We repeat the Experiment 5.2 for the rotation-augmented CIFAR10 dataset to check if LieGG retrieves symmetries from the dataset, which is more complex than the rotation-MNIST.

Revised: We employ the 7-layer perceptron with a hidden dimension equal to 400 .

Revised: We train the model forepochs on the CIFAR10 dataset augmented with random rotations.

Revised: We use Adam optimizer with the learning rate of 10 − 3 .

Revised: We select the best epoch model on the validation subset and visualize the symmetrical properties retrieved in Figure 8.

Revised: Similar to the rotation-MNIST, LieGG retrieves the correct symmetry as indicated by zeroing symmetry bias.

Revised: A.3.4 Symmetries in networks with different configurations

Revised: To evaluate networks with various configurations (width, depth, number of parameters), we consider the following family of architectures: (784 × hdim ) → p ∗ ( hdim × hdim ) → ( hdim × 10) , where p stands for the number of hidden layers and hdim is the dimension of a hidden layer.

Revised: Given the required number of parameters and the number of hidden layers, we can calculate the hidden dimension of the network.

Revised: By varying the number of parameters and the number of hidden layers we create wide and deep networks.

Revised: We follow the same procedure as in A.3.2 to train the models and extract symmetries.

Revised: A.4 LieGG implementation

Original: mental results (either in the supplemental material or as a URL)?
Revised: We provide the PyTorch implementation of the network polarization matrix computation used in the synthetic regression and the rotation-MNIST experiments. Symmetries are retrieved by performing a singular-value decomposition of the network polarization matrix as described in Section 4. ▶ synthetic regression: def polarization_matrix(model, data, dim = 5): # data: torch. FloatTensor(B, 2*dim)

Original: [No]
Revised: B = data.shape[0] data.requires_grad = True data.retain_grad() # compute network grads model.eval() y_pred= model(data) y_pred.backward(gradient=torch.ones_like(y_pred))

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
Revised: # get grads and data per input dimension dF_1 = data.grad[...,:dim].view(B, dim, 1) data_1 = data[... ,:dim].view(B, 1, dim) dF_2 = data.grad[...,dim:].view(B, dim, 1)

Original: [Yes]
Revised: data_2 =

Original: (c) Did you report error bars (e
Revised: data[...,dim:].view(B, 1, dim) # collect into the network polarization matrix C = torch.bmm(dF_1, data_1) + torch.bmm(dF_2, data_2) return C ▶ rotation-

Original: g., withrespect to the random seed after running experiments multiple times)?
Revised: : def polarization_matrix_R2(model, data): # LieGG implementation with the groups acting on R^ # data: torch.

Original: [Yes]
Revised: FloatTensor(B, 28, 28) B, H, W = data.shape # compute image grads data_grad_x = data[:, 1:, :-1] - data [:, :-1, :-1] data_grad_y =

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes]
Revised: data [:, :-1, 1:] - data[:, :-1, :-1] dI = torch.stack([data_grad_x, data_grad_y], -1) # compute network grads data =

Original: If you are using existing assets (e
Revised: data.reshape(B, -1)

Original: g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: data.requires_grad = True data.retain_grad() output = model(data) output.backward(torch.ones_like(output)) dF =

Original: [Yes] (b) Did you mention the license of the assets?
Revised: data.grad.reshape(B, H, W) dF = dF[:, :-1, :-1]

Original: [No]
Revised: # coordinate mask

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: xy = torch.meshgrid(torch.arange(0, H-1), torch.arange(0, H-1))

Original: [No]
Revised: xy =

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: torch.stack(xy, -1)

Original: using/curating? [N/A]
Revised: xy = xy / (H // 2) -

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] If you used crowdsourcing or conducted research with human subjects...
Revised: # collect into the network polarization matrix

Original: (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: C = dF[...,

Original: Board (IRB) approvals, if applicable? [N/A]
Revised: None, None] * dI[..., None] * xy[None, :, :, None, :]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
Revised: C = C.view(B, -1, 2, 2).sum(1) return C