Original: In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?
Revised: In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: It is often said that a deep learning model is “invariant” to some speciﬁc type oftransformation.
Revised: It is often said that a deep learning model is “invariant” to some speciﬁc type oftransformation.

Original: However, what is meant by this statement strongly depends onthe context in which it is made.
Revised: However, what is meant by this statement strongly depends onthe context in which it is made.

Original: In this paper we explore the nature of invarianceand equivariance of deep learning models with the goal of better understandingthe ways in which they actually capture these concepts on a formal level.
Revised: In this paper we explore the nature of invarianceand equivariance of deep learning models with the goal of better understandingthe ways in which they actually capture these concepts on a formal level.

Original: Weintroduce a family of invariance and equivariance metrics that allows us to quantifythese properties in a way that disentangles them from other metrics such as loss oraccuracy.
Revised: Weintroduce a family of invariance and equivariance metrics that allows us to quantifythese properties in a way that disentangles them from other metrics such as loss oraccuracy.

Original: We use our metrics to better understand the two most popular methodsused to build invariance into networks: data augmentation and equivariant layers.
Revised: We use our metrics to better understand the two most popular methodsused to build invariance into networks: data augmentation and equivariant layers.

Original: We draw a range of conclusions about invariance and equivariance in deep learningmodels, ranging from whether initializing a model with pretrained weights has aneffect on a trained model’s invariance, to the extent to which invariance learned viatraining can generalize to out-of-distribution data.
Revised: We draw a range of conclusions about invariance and equivariance in deep learningmodels, ranging from whether initializing a model with pretrained weights has aneffect on a trained model’s invariance, to the extent to which invariance learned viatraining can generalize to out-of-distribution data.

Original:  Introduction
Revised:  Introduction

Original: The notions of invariance and equivariance have been guiding concepts across a diverse range ofscientiﬁc domains, from physics to psychology.
Revised: The notions of invariance and equivariance have been guiding concepts across a diverse range ofscientiﬁc domains, from physics to psychology.

Original: In machine learning (ML) the concept of invarianceis frequently invoked to describe models whose output does not change when its input is transformedin ways that are irrelevant to the task the model was designed for.
Revised: In machine learning (ML) the concept of invarianceis frequently invoked to describe models whose output does not change when its input is transformedin ways that are irrelevant to the task the model was designed for.

Original: For a dog or cat image classiﬁer,for example, it is desirable for the model to be reﬂection invariant so that the same prediction is madewhether or not the input image is reﬂected across its vertical axis.
Revised: For a dog or cat image classiﬁer,for example, it is desirable for the model to be reﬂection invariant so that the same prediction is madewhether or not the input image is reﬂected across its vertical axis.

Original: This type of invariance is useful inthis model because although an image generally changes when reﬂected, whether it contains a dogor cat does not.
Revised: This type of invariance is useful inthis model because although an image generally changes when reﬂected, whether it contains a dogor cat does not.

Original: Equivariance is used to describe models whose output changes in a manner that isaligned with the way that input is transformed.
Revised: Equivariance is used to describe models whose output changes in a manner that isaligned with the way that input is transformed.

Original: For example, as an image is rotated, the position ofbounding boxes predicted by an object detector should also rotate (thus, the object detector should berotation equivariant).
Revised: For example, as an image is rotated, the position ofbounding boxes predicted by an object detector should also rotate (thus, the object detector should berotation equivariant).

Original: While mathematicians have developed rigorous theory that can describe invariance and equivariance,the amount of this that is actually used in the context of ML varies dramatically among differentworks.
Revised: While mathematicians have developed rigorous theory that can describe invariance and equivariance,the amount of this that is actually used in the context of ML varies dramatically among differentworks.

Original: Within computer vision, research on relatively simple image transformations (i.e., rotationsor translations) is often presented within a solid group-theoretic framework [4; 44; 5].
Revised: Within computer vision, research on relatively simple image transformations (i.e., rotationsor translations) is often presented within a solid group-theoretic framework [5; 53; 8].

Original: Other workdealing with more complex types of transformations such as changes in image background have (bynecessity) needed to be more informal [43].
Revised: Other workdealing with more complex types of transformations such as changes in image background have (bynecessity) needed to be more informal [52].

Original: Furthermore, while model invariance and equivarianceare frequently a central component in a broad range of ML research, limited effort has been putinto trying to measure them directly.
Revised: Furthermore, while model invariance and equivarianceare frequently a central component in a broad range of ML research, limited effort has been putinto trying to measure them directly with the purpose of understanding the general invariance andequivariance properties of deep learning models.

Original: Rather, most works measure them indirectly through other ML metrics that align with the ultimate purpose of the model (e.g., loss or accuracy).
Revised: Rather, most works measure them indirectly throughother ML metrics that align with the ultimate purpose of the model (e.g., loss or accuracy).

Original: While sucha strategy makes sense when optimizing model performance is the primary objective, we miss anopportunity to better understand how and why deep neural networks work (or fail).
Revised: Whilesuch a strategy makes sense when optimizing model performance is the primary objective, we missan opportunity to better understand how and why deep neural networks work (or fail).

Original: The purpose of this paper is to propose a group-theoretic family of metrics, associated to an arbitrary group of symmetries G , which we call G -empirical equivariance deviation ( G -EED), that quantify the
Revised: The purpose of this paper is to propose a group-theoretic family of metrics, associated to an arbitrary groupof symmetries G , which we call G -empirical equivariance deviation ( G -EED), that quantify the

Original: G -equivariance (and G -invariance) of a model.
Revised: G -equivariance (and G -invariance) of a model.

Original: Informally, the G -EED of a model measures the extent to which it fails to be equivariant on a speciﬁcdata distribution and with respect to a speciﬁc notion of distance in output space.
Revised: Informally, the G -EED of a model measures the extent to which it fails to be equivariant on a speciﬁcdata distribution and with respect to a speciﬁc notion of distance in output space.

Original: This aligns withneeds in ML where the user of a model may care only that their model is equivariant on data that themodel will actually encounter in practice and not on any possible input.
Revised: This aligns withneeds in ML where the user of a model may care only that their model is equivariant on data that themodel will actually encounter in practice and not on any possible input.

Original: Further, since invariance is aspecial case of equivariance, G -EED also measures the extent to which a model fails to be invariantto the action of G .
Revised: Further, since invariance is aspecial case of equivariance, G -EED also measures the extent to which a model fails to be invariantto the action of G .

Original: To show the breadth of the G -EED concept, we give a number of different ways itcan be applied to measure different aspects of equivariance in a model.
Revised: To show the breadth of the G -EED concept, we give a number of different ways itcan be applied to measure different aspects of equivariance in a model.

Original: For example, we show how
Revised: For example, we show how

Original: G -EED can be applied to a model’s latent space representations to measure the extent to which amodel extracts G -invariant features.
Revised: G -EED can be applied to a model’s latent space representations to measure the extent to which amodel extracts G -invariant features.

Original: Finally, we use G -EED to answer a range of questions about invariance and equivariance in neuralnetworks, with a focus on the two most popular ways of inducing invariance in these models: dataaugmentation and equivariant architectures.
Revised: Finally, we use G -EED to answer a range of questions about invariance and equivariance in neuralnetworks, with a focus on the two most popular ways of inducing invariance in these models: dataaugmentation and equivariant architectures.

Original: Some of the conclusions we draw from our experimentsinclude the following.
Revised: Some of the conclusions we draw from our experimentsinclude the following.

Original: (1) Training with augmentation does not tend to induce invariance throughlearned equivariant layers; rather, invariance arises through some other mechanism.
Revised: (1) Training with augmentation does not tend to induce invariance throughlearned equivariant layers; rather, invariance arises through some other mechanism.

Original: (2) Invariancelearned through augmentation does generalize mildly to out-of-distribution data (e.g., commonimage corruptions), but apparent invariance seen for far out-of-distribution data (e.g., images from acompletely different domain) is probably a result of model insensitivity.
Revised: (2) Invariancelearned through augmentation does generalize mildly to out-of-distribution data (e.g., commonimage corruptions), but apparent invariance seen for far out-of-distribution data (e.g., images froma completely different domain) could be the result of model insensitivity.

Original: (3) Invariance should not beassumed to correlate with model performance: models with random weights can be more invariantthan models with learned or hard-coded invariance.
Revised: (3) Invariance shouldnot be assumed to correlate with model performance: models with random weights can be moreinvariant (in the usual mathematical sense) than models with learned or hard-coded invariance.

Original: (4) Models initialized with pretrained weightstend to have different invariance or equivariance properties than models initialized with randomweights. Although the models and pretrained weights that we tested extracted fewer invariant featureson average, their ﬁnal predictions exhibited similar levels of invariance when compared to modelsinitialized with random weights.
Revised: Theauthors propose that the ML community needs to identify new notions that describe invariance thatspeciﬁcally captures ideas of invariance that are relevant to ML.(4) Models initialized with pretrainedweights tend to have different invariance or equivariance properties than models initialized withrandom weights though whether these models are judged to be more invariant depends on speciﬁcnotion of distance one chooses to use.

Original: In summary, the contributions of this paper include:
Revised: In summary, the contributions of this paper include:

Original: • The introduction of G-empirical equivariance deviation, the ﬁrst family of metrics that canrigorously and directly measure invariance and equivariance in deep learning models.•
Revised: • The introduction of G-empirical equivariance deviation, the ﬁrst family of metrics that can rigorously and directly measure invariance and equivariance in deep learning models. • A demonstration of the ﬂexibility of G-EED, showing that it can be applied easily to a range of different components of a deep learning model

Original: A demonstration of the ﬂexibility of G-EED, showing that it can be applied easily to a rangeof different components of a deep learning model to measure different forms of invarianceand equivariance.• Use of G-EED to answer a range of questions, shedding light on the extent to which neuralnetworks are or are not invariant and equivariant.
Revised: to measure different forms of invariance and equivariance. • Use of G-EED to answer a range of questions, shedding light on the extent to which neural networks are or are not invariant and equivariant.

Original:  Related Work
Revised:  Related Work

Original: The literature on invariance and equivariance in machine learning can roughly be partitioned intotwo groups: those works that focus on how to build invariance and equivariance into a model or itscomponents [21; 5; 44; 38; 6; 39; 15; 8; 31; 36; 1] and those works that focus on the theory behindinvariance or equivariance [2;  4; 3; 28; 35].
Revised: The literature on invariance and equivariance in machine learning can roughly be partitioned intotwo groups: those works that focus on how to build invariance and equivariance into a model or itscomponents [27; 8; 53; 46; 9; 47; 19; 11; 37; 42; 1] and those works that focus on the theory behindinvariance or equivariance [2; 5; 4; 34; 41].

Original: Three common approaches are used to build invariance into deep learning models: data augmentation,feature averaging, and equivariant architectures.
Revised: Three common approaches are used to build invariance into deep learning models: data augmentation,feature averaging, and equivariant architectures.

Original: In this paper we focus on the ﬁrst and third ofthese.
Revised: In this paper we focus on the ﬁrst and third ofthese.

Original: Outside of a few ubiquitous layer types (such as standard translation equivariant convolutionallayers), data augmentation is by far the most commonly used method, being a standard component ofmany training routines, particularly in computer vision.
Revised: Outside of a few ubiquitous layer types (such as standard translation equivariant convolutionallayers), data augmentation is by far the most commonly used method, being a standard component ofmany training routines, particularly in computer vision.

Original: Although the details differ between differentapproaches, the basic idea is to randomly transform training examples as they are fed to the model,thereby encouraging the model to give the same prediction for different transformations of the sameinput [32].
Revised: Although the details differ between differentapproaches, the basic idea is to randomly transform training examples as they are fed to the model,thereby encouraging the model to give the same prediction for different transformations of the sameinput [38].

Original: Since it has become a common procedure when training deep learning models, dataaugmentation research has expanded in many directions [8; 31; 36; 1].
Revised: Since it has become a common procedure when training deep learning models, dataaugmentation research has expanded in many directions [11; 37; 42; 1].

Original: The idea that G -invariance can be hard coded into a model by combining multiple G -equivariantlayers with data reduction layers (e.g., pooling) has a long history in deep learning.
Revised: The idea that G -invariance can be hard coded into a model by combining multiple G -equivariantlayers with data reduction layers (e.g., pooling) has a long history in deep learning.

Original: The most famousexample of this idea is the conventional convolutional neural network (CNN) [26].
Revised: The most famousexample of this idea is the conventional convolutional neural network (CNN) [32].

Original: Since then amultitude of other group equivariant layers have been designed including two-dimensional rotationequivariant layers [40; 42; 30; 33; 34], three-dimensional rotational equivariant layers [39; 7; 12], layers equivariant to the Euclidean group and its subgroups [38] (which we test against in this paper),and layers that are equivariant with respect to the symmetric group [27].
Revised: Since then amultitude of other group equivariant layers have been designed including two-dimensional rotation equivariant layers [48; 51; 36; 39; 40], three-dimensional rotational equivariant layers [47; 10; 15],layers equivariant to the Euclidean group and its subgroups [46] (which we test against in this paper),and layers that are equivariant with respect to the symmetric group [33].

Original: Our work is not the ﬁrst to analyze various aspects of invariance in neural network models.
Revised: Our work is not the ﬁrst to analyze various aspects of invariance in neural network models.

Original: Lyleet al.
Revised: Lyleet al.

Original: [28] analyzed invariance with respect to the beneﬁts and limitations of data augmentation andfeature averaging, presenting both theoretical and empirical arguments for using feature averagingover data augmentation.
Revised: [34] analyzed invariance with respect to the beneﬁts and limitations of data augmentation andfeature averaging, presenting both theoretical and empirical arguments for using feature averagingover data augmentation.

Original: More recently, Chen et al.
Revised: More recently, Chen et al.

Original: [4] presented a useful group-theoretic frameworkwith which to understand data augmentation.
Revised: [5] presented a useful group-theoretic frameworkwith which to understand data augmentation.

Original: Relevant to the present work, Chen et al.
Revised: Relevant to the present work, Chen et al.

Original: [3] introduceda notion of approximate invariance .
Revised: [4] introduceda notion of approximate invariance .

Original: Unlike that work, however, which focused on theoretical resultsrelated to data augmentation, this paper aims to introduce metrics that can be applied to modern deeplearning architectures and answers questions about invariance from an empirical perspective. 
Revised: Unlike that work, however, which focused on theoretical resultsrelated to data augmentation, this paper aims to introduce metrics that can be applied to modern deeplearning architectures and answers questions about invariance from an empirical perspective. Thereare a number of existing works that proposed metrics aimed at measuring the extent to which a modelis not equivariant (e.g. [7; 22; 17; 44; 50]).

Revised: Our work differs from these in two ways: (1) we buildgeneral metrics based on basic group theory that are designed to work across groups and datatypesand (2) unlike other works that use their metric to evaluate the equivariance of a speciﬁc model, weuse our metrics to explore how models learn (or do not learn) to be equivariant generally.

Original: Finally, a range of recent works have shown that even beyond the standard evaluation statistics (e.g.,accuracy), invariance is an important concept to consider when studying deep learning models.
Revised: Finally, a range of recent works have shown that even beyond the standard evaluation statistics (e.g.,accuracy), invariance is an important concept to consider when studying deep learning models.

Original: Forexample, Kaur et al.
Revised: Forexample, Kaur et al.

Original: [20] showed that lack of invariance can be used to identify out-of-distributioninputs.
Revised: [25] showed that lack of invariance can be used to identify out-of-distributioninputs.

Original: A further series of works investigated whether excessive invariance can reduce adversarialrobustness [18; 19; 35].
Revised: A further series of works investigated whether excessive invariance can reduce adversarialrobustness [23; 24; 41].

Original: All of this work reinforces one of the primary messages of this paper, that itis important to be able to measure invariance and equivariance directly in a model.
Revised: All of this work reinforces one of the primary messages of this paper, that itis important to be able to measure invariance and equivariance directly in a model.

Original:  Quantifying Invariance and Equivariance
Revised:  Quantifying Invariance and Equivariance

Original: We begin this section by recalling the mathematical deﬁnitions of equivariance and invariance.
Revised: We begin this section by recalling the mathematical deﬁnitions of equivariance and invariance.

Original: Wepresent these deﬁnitions in terms of the mathematical concept of a group, which formally capturesthe notion of symmetry [11].
Revised: Wepresent these deﬁnitions in terms of the mathematical concept of a group, which formally capturesthe notion of symmetry [14].

Original: Assume that G is a group.
Revised: Assume that G is a group.

Original: We say that G acts on sets X and Y if there are maps  X : G ⇥ X!
Revised: We say that G acts on sets X and Y if there are maps φ X : G × X

Original:  Xand  Y : G ⇥ Y ! Y that respect the composition operation of G .
Revised: → Xand φ Y : G × Y → Y that respect the composition operation of G .

Original: That is, for g 1 , g 2 2 G and 111 x X ,
Revised: That is, for g 1 , g 2 ∈ G andx ∈ X ,

Original: [Equation]
Revised: [Equation]

Original: with an analogous condition for  Y .
Revised: with an analogous condition for φ Y .

Original: Whenever the meaning is clear, we simplify notation bywriting  X ( g, x ) =
Revised: Whenever the meaning is clear, we simplify notation bywriting φ X ( g, x ) =

Original: gx (with an analogous convention for  Y ).
Revised: gx (with an analogous convention for φ Y ).

Original: A map f : X ! Y is said to be
Revised: A map f : X → Y is said to be

Original: G - equivariant if for all x 2 X and g 2 G ,
Revised: G - equivariant if for all x ∈ X and g ∈ G ,

Original: [Equation]
Revised: [Equation]

Original: In the case where the map  Y is trivial so that gy = y for all g 2 G and y 2 Y , we say that f is
Revised: In the case where the map φ Y is trivial so that gy = y for all g ∈ G and y ∈ Y , we say that f is

Original: G - invariant .
Revised: G - invariant .

Original: Thus, invariance is a special case of equivariance.
Revised: Thus, invariance is a special case of equivariance.

Original: Assume that f : X ! Y is a neural network where X is the ambient space of input data and Y isthe target space.
Revised: Assume that f : X → Y is a neural network where X is the ambient space of input data and Y isthe target space.

Original: In many cases there is a natural way to factorize f into a composition f= f 2  f 1where f 1 : X !
Revised: In many cases there is a natural way to factorize f into a composition f

Original:  Z is known as the feature extractor, Z is the latent space of f , and f 2 : Z ! Yis the classiﬁer.
Revised: = f 2 ◦ f 1where f 1 : X → Z is known as the feature extractor, Z is the latent space of f , and f 2 : Z → Yis the classiﬁer.

Original: For example, if f is a ResNet50 CNN [16] then f 1 may consist of all residualblocks while f 2 would consist of the ﬁnal afﬁne classiﬁcation and softmax layers.
Revised: For example, if f is a ResNet50 CNN [20] then f 1 may consist of all residualblocks while f 2 would consist of the ﬁnal afﬁne classiﬁcation and softmax layers.

Original: We say thatmachine learning model
Revised: We say thatmachine learning model

Original: f
Revised: f

Original: extracts G -equivariant features if f 1 is a G -equivariant function.
Revised: extracts G -equivariant features if f 1 is a G -equivariant function.

Original: This is anespecially meaningful distinction in the context of transfer learning where invariance or equivariancecan be transferred to a new task via the invariance or equivariance of f 1 .
Revised: This is anespecially meaningful distinction in the context of transfer learning where invariance or equivariancecan be transferred to a new task via the invariance or equivariance of f 1 .

Original: Note that the deﬁnition of
Revised: Note that the deﬁnition of

Original: G -equivariant feature extraction requires a well-deﬁned action of G on Z , which may not be obviousin many cases.
Revised: G -equivariant feature extraction requires a well-deﬁned action of G on Z , which may not be obviousin many cases.

Original: Because the trivial action is deﬁned for any G and Z , we can always ask whether fextracts G -invariant features .
Revised: Because the trivial action is deﬁned for any G and Z , we can always ask whether fextracts G -invariant features .

Original: The following proposition provides some insight into how the invariance (or lack of invariance) of f 1relates to the invariance (or lack of invariance) of f .
Revised: The following proposition provides some insight into how the invariance (or lack of invariance) of f 1relates to the invariance (or lack of invariance) of f .

Original: Proposition 3.1.
Revised: Proposition 3.1.

Original: Let f : X ! Y be a function that decomposes into f = f 2  f 1 where f 1 : X ! Zand f 2 : Z !
Revised: Let f : X → Y be a function that decomposes into f = f 2 ◦ f 1 where f 1 : X → Zand f 2 : Z → Y .

Original: Y . Suppose that G acts on X , Z , and Y .
Revised: Suppose that G acts on X , Z , and Y .

Original: f can be G -invariant even if f 1 and f 2 are not.
Revised: f can be G -invariant even if f 1 and f 2 are not.

Original: If f 1 is G -invariant, than f is G -invariant.
Revised: If f 1 is G -invariant, then f is G -invariant.

Original: A proof of Proposition 3.1 can be found in Section A.3.
Revised: A proof of Proposition 3.1 can be found in Section A.6.

Original: Note that Proposition 3.1.2 implies that ifearlier layers of a network achieve invariance with respect to some group, then this invariance willpersist into later layers.
Revised: Note that Proposition 3.1.2 implies that ifearlier layers of a network achieve invariance with respect to some group, then this invariance willpersist into later layers.

Original: This may be seen as part of the justiﬁcation of the common practice within the equivariant architectures community of building invariance through successive combinations of
Revised: This may be seen as part of the justiﬁcation of the common practice withinthe equivariant architectures community of building invariance through successive combinations of

Original: G -equivariant layers and pooling layers.
Revised: G -equivariant layers and pooling layers.

Original: Of course, the statement holds only when exact invariance isachieved.
Revised: Of course, the statement holds only when exact invariance isachieved.

Original: We see below that this is not generally the case.
Revised: We see below that this is not generally the case.

Original: 3.1 Measuring equivariance
Revised: 3.1 Measuring equivariance

Original: In this section we assume that both X and Y are vector spaces and the action of G on both X and  Y is linear.
Revised: In this section we assume that both X and Y are vector spaces and the action of G on both X and Yis linear. In all of our experiments we assume that the action of G on X and Y is known.

Original: In the case that f is also a linear map, the equivariance of f can be checked directly bychecking equivariance on a basis for X .
Revised: In the casethat f is also a linear map, the equivariance of f can be checked directly by checking equivariance ona basis for X .

Original: By extension, the equivariance of many common types ofneural network layers can be checked when these can be framed as linear maps (e.g., convolutionallayers).
Revised: By extension, the equivariance of many common types of neural network layers can bechecked when these can be framed as linear maps (e.g., convolutional layers).

Original: However, there is no systematic procedure for checking the equivariance of a nonlinearfunction f .
Revised: However, there is nosystematic procedure for checking the equivariance of a nonlinear function f .

Original: If f is the composition of a sequence of functions f 1 , f 2 , . . .
Revised: If f is the compositionof a sequence of functions f 1 , f 2 , . . .

Original: , f n and we can check thateach is equivariant, then we know that f is equivariant, but we cannot prove that a function is notequivariant just by proving that each layer is not equivariant (this follows from Proposition 3.1.1).
Revised: , f n and we can check that each is equivariant, then we knowthat f is equivariant, but we cannot prove that a function is not equivariant just by proving that eachlayer is not equivariant (this follows from Proposition 3.1.1).

Original: We are thus motivated to introduce a family of metrics that can be used to empirically measure theextent to which a function deviates from being equivariant on a data distribution D on X .
Revised: We are thus motivated to introduce afamily of metrics that can be used to empirically measure the extent to which a function deviatesfrom being equivariant on a data distribution D on X .

Original: Since we assume in this section that the action of G on Y is linear, we can deﬁne the kernel of thisaction, ker(  Y ) , which is a subgroup of G .
Revised: Since we assume in this section that the action of G on Y is linear, we can deﬁne the kernel of thisaction, ker( φ Y ) , which is a subgroup of G .

Original: ker(  Y ) consists of all those g 2 G such that g acts asthe identity on Y .
Revised: ker( φ Y ) consists of all those g ∈ G such that g acts asthe identity on Y .

Original: For x 2 X , we deﬁne ˆ f ( x ) to be the expected value of f ( gx ) over ker(  Y ) ,
Revised: For x ∈ X , we deﬁne ˆ f ( x ) to be the expected value of f ( gx ) over ker( φ Y ) ,

Original: [Equation]
Revised: [Equation]

Original: where µ is the usual normalized Haar measure on subgroup ker( G ) .
Revised: where µ is the usual normalized Haar measure on subgroup ker( G ) .

Original: Note that when f is G -equivariant,then for each g 2 ker(  Y ) , f ( gx ) =
Revised: Note that when f is G -equivariant,then for each g ∈ ker( φ Y ) , f ( gx ) =

Original: gf ( x ) =
Revised: gf ( x ) =

Original: f
Revised: f

Original: ( x ) and hence ˆ f = f .
Revised: ( x ) and hence ˆ f = f .

Original: Let ( D , ⌫ ) be a probability distribution on X and let m :Y ⇥ Y !
Revised: Let ( D , ν ) be a probability distribution on X and let m :

Original:  R  0 be a distance function on Y .
Revised: Y × Y → R ≥ 0 be a distance function on Y .

Original: 157 We can use m to measure the extent to which f deviates from being G -equivariant by computing
Revised: We can use m to measure the extent to which f deviates from being G -equivariant by computing

Original: [Equation]
Revised: [Equation]

Original: where this time µ is the normalized Haar measure on G .
Revised: where this time µ is the normalized Haar measure on G .

Original: Note that the argument m ( f ( gx ) , g ˆ f ( x ))measures the extent to which (1) fails to hold across distribution D , except that gf ( x ) is replacedby g ˆ f ( x ) .
Revised: Note that the argument m ( f ( gx ) , g ˆ f ( x ))measures the extent to which (1) fails to hold across distribution D , except that gf ( x ) is replaced byg ˆ f ( x ) .

Original: We use ˆ f ( x ) because it averages over all values in the orbit of x (under G ) that shouldmap to f ( x ) , were f  genuinely G -equivariant.
Revised: We use ˆ f ( x ) because it averages over all values in the orbit of x (under G ) that should map tof ( x ) . If f were genuinely G -equivariant, all these values f ( gx ) would yield f ( x ) .

Original: This choice is supported by the fact that it naturallyinterpolates between the two extreme cases: G acts trivially on Y (invariance) and G acts faithfullyon Y .
Revised: This choice issupported by the fact that it naturally interpolates between the two extreme cases: G acts trivially on Y (invariance) and G acts faithfully on Y .

Original: In the former case ˆ f ( x ) is the average value of f ( gx ) over all of g 2 G and in the latter caseˆ f ( x ) =
Revised: In the former case ˆ f ( x ) is the average value of f ( gx ) overall of g ∈ G and in the latter case ˆ f ( x ) =

Original: f
Revised: f

Original: ( x ) .
Revised: ( x ) .

Original: Proposition 3.2.
Revised: Proposition 3.2.

Original: Let f : X ! Y be a continuous function, G a group that acts linearly on both Xand Y , and m :Y ⇥ Y !
Revised: Let f : X → Y be a continuous function, G a group that acts linearly and faithfullyon both X and Y , and m :

Original:  R  0 a metric.
Revised: Y × Y → R ≥ 0 a metric.

Original: Let D be a distribution on X .
Revised: Let D be a distribution on X .

Original: Then (3) is zero  ifand onlyif f is G -equivariant almost surely, i.e., on a set of measure 1 .
Revised: Then (3) is zero ifand only if f is G -equivariant almost surely, i.e., on a set of measure 1 .

Original: We provide a proof of Proposition 3.2 in Appendix A.

Revised: We provide a proof of Proposition 3.2 in Appendix A.

Original: To approximate (3) for real models and data where we always work with ﬁnite groups and ﬁnitesamples of D , we deﬁne the G -EED of f with respect to m to be
Revised: To approximate (3) for real models and data where we always work with ﬁnite groups and ﬁnitesamples of D , we deﬁne the G -EED of f with respect to m to be

Original: [Equation]
Revised: [Equation]

Original: Note that since G is a ﬁnite group (and hence discrete), the Haar measure turns into the usual countingmeasure.
Revised: Note that since G is a ﬁnite group (and hence discrete), the Haar measure turns into the usual countingmeasure.

Original: In the remainder of this section we describe some speciﬁc types of G -EED that may be relevant tocomputer vision tasks.
Revised: In the remainder of this section we describe some speciﬁc types of G -EED that may be relevant tocomputer vision tasks.

Original: For channelwise G -EED, large values indicate more equivariance; for latentspace G -EED and softmax G -EED, larger values  indicate less invariance.
Revised: By convention, when using a distance function m for which larger values ofm ( x 1 , x 2 ) indicates that x 1 and x 2 are “closer” (the opposite of a proper metric), we attach a negativesign to m . For example, rather than using cosine similarity, we use negative cosine similarity (e.g., 5). This way, larger values of G -EED consisently indicate less invariance, regardless of the m used.

Original: The diagram in Figure 1provides a summary of all of these.
Revised: Thediagram in Figure 1 provides a summary of the types of G -EED studied in this paper. we investigate

Original: The channelwise G -equivariance of convolutional layer activations: Throughout most layers of a
Revised: The channelwise G -equivariance of convolutional layer activations: Throughout most layers of a

Original: CNN an individual image is represented as a 3 -tensor.
Revised: CNN an individual image is represented as a 3 -tensor.

Original: Let f ` :X !
Revised: Let f (cid:96) :

Original:  R C ` ⇥ H ` ⇥ W ` be the composition of the ﬁrst ` layers of a CNN such that for input image x 2 X , f ` ( x ) is a C ` ⇥ H ` ⇥ W ` tensorwhere the ﬁrst dimension corresponds to the channels of the representation and the second and thirddimensions correspond to the two spatial dimensions.
Revised: X → R C (cid:96) × H (cid:96) × W (cid:96) be the compositionof the ﬁrst (cid:96) layers of a CNN such that for input image x ∈ X , f (cid:96) ( x ) is a C (cid:96) × H (cid:96) × W (cid:96) tensorwhere the ﬁrst dimension corresponds to the channels of the representation and the second and thirddimensions correspond to the two spatial dimensions.

Original: Suppose that G is a ﬁnite group that acts on images and other 2 -tensors (e.g., rotations, translations,and reﬂections).
Revised: Suppose that G is a ﬁnite group that acts on images and other 2 -tensors (e.g., rotations, translations,and reﬂections).

Original: To understand the extent to which the ﬁrst ` layers of the network are G -equivariant,we can measure the G -EED of f ` .
Revised: To understand the extent to which the ﬁrst (cid:96) layers of the network are G -equivariant,we can measure the G -EED of f (cid:96) .

Original: Although there are numerous choices of m for (3) that could beused to measure the channelwise difference between f ` ( gx ) and gf ` ( x ) , we choose the following: let 
Revised: Although there are numerous choices of m for (3) that could beused to measure the channelwise difference between f (cid:96) ( gx ) and gf (cid:96) ( x ) , we choose the following: let [Equation]

Original: S : R H ` ⇥ W ` ⇥ R H ` ⇥ W ` !
Revised: × W (cid:96) →

Original: [0 , 1] be the cosine similarity on individual channels treated as vectors in  R H ` W ` .
Revised: [0 , 1] be the cosine similarity on individual channels treated as vectors in 197 R H (cid:96) W (cid:96) .

Original: Write [ f ` ( x )] i for the i th channel of f ` ( x ) .
Revised: Write [ f (cid:96) ( x )] i for the i th channel of f (cid:96) ( x ) .

Original: Then we set
Revised: Then we set

Original: [Equation]
Revised: [Equation]

Original: That is, let m be the  average cosine similarity between individual channels of f ` ( gx ) and g ˆ f ` ( x ) . 
Revised: That is, let m be the negative of the average cosine similarity between individual channels of f (cid:96) ( gx )and g ˆ f (cid:96) ( x ) . This gives:

Original:  Note that since rotation, translation, and reﬂection groups all act faithfully on R H ` ⇥ W ` , for thesespeciﬁc G , ˆ f = f .
Revised: [Equation] Note that since rotation, translation, and reﬂection groups all act faithfully on R H (cid:96) × W (cid:96) , for thesespeciﬁc G , ˆ f = f .

Original: We call this version of G -EED channelwise G -EED .
Revised: We call this version of G -EED channelwise G -EED .

Revised: This metric assumes that Gacts on each channel in a 3-tensor independently.

Revised: It does not account for the more complicated settingwhere the action of G either permutes or mixes channels of R C (cid:96) × H (cid:96) × W (cid:96) in a non-trivial way.

Revised: In thiswork we always assume that the action of G on channels of R C (cid:96) × H (cid:96) × W (cid:96) is identical as the actionof G on X (up to differences in spatial scale).

Revised: So if G = C 8 and G rotates the spatial dimension ofinput images, we assume G rotates each individual channel of hidden representations as well.

Original: Invariance of latent space representations: Consider a model f consisting of feature extractorf 1 : X ! R k and classiﬁer f 2 : R k!
Revised: Invariance of latent space representations: Consider a model f consisting of feature extractorf 1 : X → R k and classiﬁer f 2 : R k

Original:  R n .
Revised: → R n .

Original: If G is some group that acts on X , we can ask whetherf 1 extracts G -invariant features.
Revised: If G is some group that acts on X , we can ask whetherf 1 extracts G -invariant features.

Original: Deﬁne M to be  the average ` 2 -distance between distinct points inthe set f 1 ( D ) .
Revised: Deﬁne M to be either the average (cid:96) 2 -distance or cosine similaritybetween distinct points in the set f 1 ( D ) .

Original: By setting m to be ` 2 -distance, we calculate the latent space G -EED as
Revised: By setting m to be the (cid:96) 2 -distance (respectively cosinesimilarity), we calculate the latent space G -EED as

Original: [Equation]
Revised: [Equation]

Original: Note that because in this setting G acts trivially on the latent space R k (because of the invarianceassumption), ˆ f 1 ( x ) = [Equation]
Revised: Note that because in this setting G acts trivially on the latent space R k (because of the invarianceassumption), ˆ f 1 ( x ) = 1 | G | (cid:80) g ∈ G f 1 ( gx ) .

Original: We describe why the normalization term M is necessary in Section A.5 of the Appendix.
Revised: We describe why the normalization term M is necessary in Section A.8 of the Appendix.

Original: See also Figure 7.
Revised: See also Figure 15.

Revised: As we show in Section 4.1, it is often useful toconsider latent space G -EED with respect to both Euclidean distance and negative cosine similarity.

Original: Invariance at the softmax layer of classiﬁers:
Revised: Invariance at the softmax layer of classiﬁers:

Original: While the two examples above focused on calculating invariance and equivariance at differentintermediate layers of computer vision models, invariance of classiﬁcation models is most commonlyconsidered with respect to the ﬁnal softmax output.
Revised: While the two examples above focused on calculating invariance and equivariance at differentintermediate layers of computer vision models, invariance of classiﬁcation models is most commonlyconsidered with respect to the ﬁnal softmax output.

Original: In this paper we choose to use KL-divergencefor m because it is a distance function designed to handle probability distributions, which softmaxoutputs approximate.
Revised: In this paper we choose to use KL-divergencefor m because it is a distance function designed to handle probability distributions, which softmaxoutputs approximate.

Original: In particular,
Revised: In particular,

Original: [Equation]
Revised: [Equation]

Original: We call this metric the softmax G -EED .
Revised: We call this metric the softmax G -EED .

Revised: Note that like all the metrics described in this section, softmax

Revised: G -EED strictly measures invariance and is frequently not in line with task performance.

Original:  Understanding invariance and equivariance in 
Revised:  Understanding invariance and equivariance in deep learning

Original: learning

Original: In this section we describe a range of experiments designed to probe the invariance and equivarianceof computer vision models.

Original: To prototype G -EED we experiment with the ﬁnite cyclic group C 8 thatacts on images by 45  rotations.
Revised: To prototype G -EED we experiment with the ﬁnite cyclic group C 8 that acts on images by 45 ◦rotations.

Original: This symmetry is relevant to datasets where imagery does not havea preferred orientation.
Revised: This symmetry is relevant to datasets where imagery does not have a preferred orientation.

Original: The datasets we use include Rotated MNIST which consists of randomlyrotated MNIST digits from classes 0-8 and xView Maritime , a chipped version of the xView objectdetector overhead dataset.
Revised: The datasets we use include Rotated MNIST which consists of randomly rotated MNIST digits fromclasses 0-8 and xView Maritime , a chipped version of the xView object detector overhead datasetcomposed of the nine maritime classes. We chose xView Maritime because it is a more real-worlddataset that has an obvious symmetry (rotation) that is not hardcoded into standard CNNs.

Original: We describe these and other datasets we use in detail in Section A.2.
Revised: Wedescribe these and other datasets we use in detail in Section A.2.

Original: To reduce the computational burden, when calculating (4) we only used 50 randomly chosen examplesfrom the corresponding dataset.
Revised: To reduce the computational burden, when calculating (4) we only used 50 randomly chosen examplesfrom the corresponding dataset.

Original: We found that evaluating at more points did not substantially changethe results.
Revised: We found that evaluating at more points did not substantially changethe results.

Original: When not otherwise stated all invariance and equivariance measurements were calculatedusing points from the test set, not the training set.
Revised: When not otherwise stated all invariance and equivariance measurements were calculatedusing points from the test set, not the training set.

Original: The normalization constant for latent space G -EEDis calculated over 200 randomly chosen points.
Revised: The normalization constant for latent space G -EEDis calculated over 200 randomly chosen points.

Original: 4.1 Do networks trained with augmentation learn equivariant layers?
Revised: 4.1 Do networks trained with augmentation learn equivariant layers?

Original: In this section we apply our metrics to standard CNNs and C 8 -rotation equivariant, steerable CNNs( C 8 -CNN) [38] to compare the invariance or equivariance of a model trained with augmentationwith the invariance or equivariance of a model that is explicitly designed to learn invariance throughprogressive equivariant layers and pooling.
Revised: In this section we apply our metrics to standard CNNs and C 8 -rotation equivariant, steerable CNNs( C 8 -CNN) [46] to compare the invariance or equivariance of a model trained with augmentationwith the invariance or equivariance of a model that is explicitly designed to learn invariance throughprogressive equivariant layers and pooling.

Original: We train 10 copies of each model type for 2,000 iterationson rotated MNIST.
Revised: We train 10 copies of each model type for 2,000 iterationson rotated MNIST.

Original: We apply random rotation augmentation to the models during training and allachieve an accuracy of over 98% on the test set.
Revised: We apply random rotation augmentation to the models during training and allachieve an accuracy of over 98% on the test set.

Original: By design, for any 1  k  6 , the ﬁrst k -blocks ofone of the C 8 -CNN models will be approximately rotation equivariant.
Revised: By design, for any 1 ≤ k ≤ 6 , the ﬁrst k -blocks ofone of the C 8 -CNN models will be approximately rotation equivariant.

Original: The standard wisdom is that,with pooling, this will lead to an invariant representation in the latent space.
Revised: The standard wisdom is that,with pooling, this will lead to an invariant representation in the latent space.

Original: Despite the fact that both families of models achieve similar performance, we note that their invarianceand equivariance properties differ signiﬁcantly.
Revised: Despite the fact that both families of models achieve similar performance, we note that their invarianceand equivariance properties differ signiﬁcantly.

Original: The channelwise C 8 -EED of the composition ofthe ﬁrst k blocks (where 1  k  6 ) of the C 8 -CNN is signiﬁcantly higher than that of the CNN,indicating that (as claimed) the C 8 -CNN is more equivariant with respect to rotation ofinput imagesof 45  (see Figure 2 for examples of output at blocks 2 (left) and 5 (right)).
Revised: The channelwise C 8 -EED of the composition ofthe ﬁrst k blocks (where 1 ≤ k ≤ 6 ) of the C 8 -CNN is signiﬁcantly higher than that of the CNN,indicating that (as claimed) the C 8 -CNN is more equivariant with respect to individual channelrotations of 45 ◦ in both input and hidden representations (see Figure 6 for examples of output atblocks 2 (left) and 5 (right)).

Original: Beyond that we see thatequivariance decays for both models as input travels through additional blocks.
Revised: Beyond that we see that equivariance decays for both models as inputtravels through additional blocks.

Original: This is particularlytrue for the C 8 -CNN.
Revised: This is particularly true for the C 8 -CNN.

Original: We conjecture that this is likely caused by the accumulation of interpolationartifacts associated with rotation.
Revised: We conjecture that this islikely caused by the accumulation of interpolation artifacts associated with rotation.

Original: Perhaps even more interesting is the fact that for no layer of eithermodel do the channelwise C 8 -EED values increase beyond a small jump at the beginning of training.
Revised: Perhaps evenmore interesting is the fact that for no layer of either model do the channelwise C 8 -EED valuesincrease beyond a small jump at the beginning of training.

Original: Indeed, for most of training, these values decay.

Original: This suggests that layerwise equivariance may notalways be aligned with the learning objective and that learned invariance does not arise throughlearned equivariance.
Revised: This suggests that layerwise equivariancemay not always be aligned with the learning objective and that learned invariance does not arisethrough the naive form of learned equivariance.

Original: Both in terms of the  latent space C 8 -EED and softmax C 8 -EED, the C 8 -CNNs show more invariancethan the conventional CNNs (see Figure 3) but the difference was less dramatic than it was forchannelwise C 8 -EED.
Revised: Both in terms of the Euclidean distance version of latent space C 8 -EED and softmax C 8 -EED, the C 8 -CNNs show more invariance than the conventional CNNs (see Figure 2) but the difference was less dramatic than it was for channelwise C 8 -EED.

Original: This suggests that the CNNs likely caught up in terms of C 8 -invarianceusing mechanisms distinct from layerwise C 8 -equivariance. 
Revised: This suggests that the CNNs likely caught up interms of C 8 -invariance using mechanisms distinct from layerwise C 8 -equivariance. Notable alsois

Original: Notable also is the fact that unlikechannelwise C 8 -EED, which did not indicate increased C 8 -equivariance over the course of training,both latent space and softmax C 8 -EED decreased with training (suggesting learned invariance). 
Revised: the fact that unlike channelwise C 8 -EED, which did not indicate increased C 8 -equivariance overthe course of training, both latent space and softmax C 8 -EED decreased with training (suggestinglearned invariance). Most surprisingly, in Figure 9 we see that the C 8 -CNNs are less invariant than the conventional CNNswith respect to the cosine similarity version of latent space C 8 -EED. We conjecture two possiblereasons for this. The ﬁrst is based on the fact that C 8 -CNNs are designed to be equivariant on thenose , not up to scaling .

Revised: The structural constraint of exact equivariance may actually be misalignedwith cosine similarity where vector magnitude no longer matters.

Original: Sowhile C 8 -equivariance is not learned by either model class, some C 8 -invariance likely is.
Revised: From this perspective, projectiverepresentation theory, where symmetries are only taken up to scaling, might be a proﬁtable directionof future research in equivariant architectures. The second possible explanation arises from Figure8 which shows that in terms of absolute cosine similarity (that is, without the normalization term M in (6)), the C 8 -CNN is more invariant. It may be that while C 8 -CNNs indeed cluster orbits ofpoints under the C 8 action closer together, the conventional CNNs are better at spreading other pointsfurther away in the latent space.

Original: Summary: Equivariant architectures and nonequivariant architectures capture invariance in distinctways.
Revised: Summary: Equivariant architectures and nonequivariant architectures capture invariance in distinctways.

Original: Nonequivariant networks do not learn layerwise equivariance as a path to invariance.
Revised: Nonequivariant networks do not learn layerwise equivariance (at least with respect to the naive

Revised: C 8 action given by rotating individual channels in a hidden representation).

Original: Ratherthey achieve their learned invariance via some other mechanism.
Revised: Rather they achieve theirlearned invariance via some other mechanism.

Original: In our experiments, equivariantarchitectures were able to deliver models that were more invariant than their nonequivariant analogboth in terms of  the features that they extracted and their ﬁnal softmax outputs.
Revised: The level of invariance measured in the latent spacedepends on the underlying metric m which is used.

Original: 4.2 Does learned invariance hold for out-of-distribution data?
Revised: 4.2 Does learned invariance hold for out-of-distribution data?

Original: It was noted in Lyle et al.
Revised: It was noted in Lyle et al.

Original: [28] that while augmentation may be satisfactory for many tasks, thereis a risk that learned invariance will not extend to out-of-distribution (OOD) data.
Revised: [34] that while augmentation may be satisfactory for many tasks, thereis a risk that learned invariance will not extend to out-of-distribution (OOD) data.

Original: We investigatethis question using G -EED metrics, studying the conventional CNN and C 8 -equivariant CNN from
Revised: We investigatethis question using G -EED metrics, studying the conventional CNN and C 8 -equivariant CNN from

Original: Section 4.1 and a conventional CNN with untrained weights that can be used for comparison.
Revised: Section 4.1 and a conventional CNN with untrained weights that can be used for comparison.

Original: Notethat because rotation invariance is at some level hard coded into the architecture of the C 8 -CNN,our default assumption is that the C 8 -CNN models will tend to be more invariant on OOD data thanmodels with learned augmentation.
Revised: Notethat because rotation invariance is at some level hard coded into the architecture of the C 8 -CNN,our default assumption is that the C 8 -CNN models will tend to be more invariant on OOD data thanmodels with learned augmentation.

Original: Images of example OOD data as well as dataset descriptions canbe found in Appendix A.2.
Revised: Images of example OOD data as well as dataset descriptions canbe found in Appendix A.2.

Original: The left plot in Figure 4 shows the latent space C 8 -EED for the three model families that we tested.
Revised: The left plot in Figure 3 shows the latent space C 8 -EED for the three model families that we tested.

Original: We can see that random weights extract signiﬁcantly less invariant latent space representationscompared to both the augmentation-trained CNNs and C 8 -CNNs.
Revised: We can see that random weights extract signiﬁcantly less invariant latent space representationscompared to both the augmentation-trained CNNs and C 8 -CNNs.

Original: Although this difference is notsurprising for the C 8 -CNNs, for the standard CNNs it indicates that augmentation does learn somerotation-invariant features that generalize to OOD data.
Revised: Although this difference is notsurprising for the C 8 -CNNs, for the standard CNNs it indicates that augmentation does learn somerotation-invariant features that generalize to OOD data.

Original: Furthermore, while C 8 -CNNs do exhibitmore latent space invariance than the augmentation-trained CNNs, the difference is insigniﬁcantcompared to the difference between augmentation-trained CNNs and CNNs with random weights.
Revised: Furthermore, while C 8 -CNNs do exhibitmore latent space invariance than the augmentation-trained CNNs, the difference is insigniﬁcantcompared to the difference between augmentation-trained CNNs and CNNs with random weights.

Original: Note that in these OOD experiments, the normalization term M in (5) is calculated using the OODdata (rather than the training or test set).
Revised: Note that in these OOD experiments, the normalization term M in (6) is calculated using the OOD data (rather than the training or test set).

Original: This choice was made when considering situations in whicha frozen network is used in combination with metric learning methods, as in Snell et al.
Revised: This choice was made when considering situations in whicha frozen network is used in combination with metric learning methods, as in Snell et al.

Original: [37], to solvea task with data drawn from a shifted distribution.
Revised: [43], to solvea task with data drawn from a shifted distribution.

Original: As a showcase of how a single model can display different types of invariance, the CNNs with randomweights are far more invariant in terms of softmax C 8 -EED than the trained C 8 -CNN, which is moreinvariant than the trained CNN (right plot, Figure 4).
Revised: As a showcase of how a single model can display different types of invariance, the CNNs with randomweights are far more invariant in terms of softmax C 8 -EED than the trained C 8 -CNN, which is moreinvariant than the trained CNN (right plot, Figure 3).

Original: We suspect that invariance of the CNNs withrandom weights arises from the fact that an untrained network tends to be insensitive to input.
Revised: We suspect that invariance of the CNNs withrandom weights arises from the fact that an untrained network tends to be insensitive to input.

Original: Thisobservation is useful to keep in mind, especially as the concept of invariance is frequently treated as a net beneﬁt to any model.
Revised: Thisobservation is useful to keep in mind, especially as the concept of invariance is frequently treated asa net beneﬁt to any model.

Original: Invariance is useful when it is aligned with other metrics more directlyrelated to the task the model was designed for.
Revised: Invariance is useful when it is aligned with other metrics more directlyrelated to the task the model was designed for.

Original: Summary: Our experiments suggest that augmentation trained networks do learn invariance thatmildly generalizes to OOD data.
Revised: Summary: Our experiments suggest that augmentation trained networks do learn invariance thatmildly generalizes to OOD data.

Original: A single models can display differing kinds of invariance (anuntrained network does not not extract invariant features but through insensitivity may make invariantpredictions).
Revised: A single models can display differing kinds of invariance (anuntrained network does not not extract invariant features but through insensitivity may make invariantpredictions).

Original: In cases where invariance is critical and OOD data is expected, equivariant architecturesmay be a safer choice than training with augmentation.
Revised: In cases where invariance is critical and OOD data is expected, equivariant architecturesmay be a safer choice than training with augmentation.

Original:  How does the use of pretrained weights affect invariance andequivariance?
Revised: 4.3 How does the use of pretrained weights affect invariance and equivariance?

Original: It has become standard practice in computer vision to initialize a model using weights trained ona large, diverse image dataset such as ImageNet [9].
Revised: It has become standard practice in computer vision to initialize a model using weights trained ona large, diverse image dataset such as ImageNet [12].

Original: It is reasonable to ask how such a strategyaffects the invariance and equivariance of the resulting models.
Revised: It is reasonable to ask how such a strategyaffects the invariance and equivariance of the resulting models.

Original: We choose four different imageclassiﬁer architectures: ResNet50 [16], AlexNet [23], DenseNet121 [17], and LeViT192 [14].
Revised: We choose four different imageclassiﬁer architectures: ResNet50 [20], AlexNet [29], DenseNet121 [21], and LeViT192 [18].

Original: Foreach of these architectures, we initialized ﬁve models with random weights and ﬁve models withweights from either the Torchvision package [29] (ResNet50, AlexNet, and DenseNet) or Timm[41] (LeViT192).
Revised: Foreach of these architectures, we initialized ﬁve models with random weights and ﬁve models withweights from either the Torchvision package [35] (ResNet50, AlexNet, and DenseNet) or Timm[49] (LeViT192).

Original: We then trained all of these models on the xView Maritime training set.
Revised: We then trained all of these models on the xView Maritime training set.

Original: Next,we evaluated the latent space C 8 -EED and softmax C 8 -EED on the xView Maritime test set.
Revised: Next,we evaluated the latent space C 8 -EED and softmax C 8 -EED on the xView Maritime test set.

Original: It isimportant to note that pretrained weights, which were generated by training on ImageNet, might notbe expected to have learned to extract fully rotationally invariant features.
Revised: It isimportant to note that pretrained weights, which were generated by training on ImageNet, might notbe expected to have learned to extract fully rotationally invariant features.

Original: Indeed, the objects foundin ImageNet mostly have a preferred orientation at which they generally are seen (there are very fewinstances of upside-down dogs or vehicles in ImageNet).
Revised: Indeed, the objects foundin ImageNet mostly have a preferred orientation at which they generally are seen (there are very fewinstances of upside-down dogs or vehicles in ImageNet).

Original: The results of our experiments are shown in Figure 5.
Revised: The results of our experiments are shown in Figure 4.

Original: Figure 5 shows the latent space C 8 -EED forall eight model types every 200 iterations.
Revised: Figure 4 shows the latent space C 8 -EED forall eight model types every 200 iterations.

Original: We see that models trained from scratch extract moreinvariant features in their latent space.
Revised: We see that models trained from scratch extract moreinvariant features in their latent space.

Original: Moreover, while the invariance of models trained from scratchtends to decrease slightly over the course of training (latent space C 8 -EED increases), the invarianceof models trained using pretrained weights tends to increase slightly over the course of training.
Revised: Moreover, while the invariance of models trained from scratchtends to decrease slightly over the course of training (latent space C 8 -EED increases), the invarianceof models trained using pretrained weights tends to increase slightly over the course of training.

Original: Thismay relate to ImageNet’s lack of object classes that appear at a wide range of orientations.
Revised: Thismay relate to ImageNet’s lack of object classes that appear at a wide range of orientations.

Original: Themodels with pretrained weights may need to learn additional rotation invariance over the course oftraining.
Revised: Themodels with pretrained weights may need to learn additional rotation invariance over the course of training.

Original: Contrast that with the weights trained from scratch that may learn rotation invariant featuresquickly which are later discarded in favor of less rotationally invariant features that better optimizethe cross-entropy loss function used to train these networks.
Revised: Contrast that with the weights trained from scratch that may learn rotation invariantfeatures quickly which are later discarded in favor of less rotationally invariant features that betteroptimize the cross-entropy loss function used to train these networks.

Original: This, along with the fact that modelsusing pretrained weights tend to have slightly higher accuracy, points to a potential tradeoff betweeninvariance and task performance.
Revised: This, along with the fact thatmodels using pretrained weights tend to have slightly higher accuracy, points to a potential tradeoffbetween invariance and task performance.

Original: Whether the robustness of the features captured by the pretrainednetworks is related to the fact that they are less invariant requires further exploration.
Revised: Whether the robustness of the features captured by thepretrained networks is related to the fact that they are less invariant requires further exploration.

Revised: Wealso calculated the latent space D 8 -EED (where D 8 is the order 16 dihedral group) for these samemodels and dataset.

Original:  On the other hand, Figure 9 in the Appendix shows that the invariance properties of pretrainednetworks vs. networks trained from stratch is somewhat more ambiguous when considering softmax
Revised: The results are recorded in Figure 10. We ﬁnd the patterns to be fairly similar. On the other hand, Figure 17 in the Appendix shows that the invariance properties of pretrainednetworks vs. networks trained from stratch is somewhat more ambiguous when considering softmax

Original: C 8 -EED.
Revised: C 8 -EED.

Original: Summary: In our experiments, models that used ImageNet pretrained weights tended to extractsomewhat less invariant features, even if the invariance of their softmax outputs tended to becomparable.
Revised: Summary: Our experiments make it clear that networks with pretrained weights incorporate invariance differently than networks trained from scratch.

Original: It may be worth considering developing large datasets for pretraining that containthe types of symmetries that are relevant to speciﬁc domains (e.g., a version of ImageNet designedfor rotation invariant imagery such as overhead imagery and microscopy).
Revised: Further research will be required to fullyunderstand these differences.

Original:  Limitations 
Revised:  Limitations and Conclusion

Original: Although the G -EED family of metrics that we introduce is ﬂexible in many ways, it is not capable ofcapturing notions of invariance that do not have a known underlying symmetry group (e.g., changesin image background).
Revised: Although the G -EED family of metrics that we introduce is ﬂexible in many ways, it is not capable ofcapturing notions of invariance that do not have a known underlying symmetry group (e.g., changesin image background), do not come from the action of a group (e.g., scaling corresponding to asemigroup action), or (for the purpose of measuring equivariance of internal representations) do nothave a known action on the latent space of a model (e.g., changes in color).

Original: An important next step would be to bring our metrics to a broader notion ofinvariance.
Revised: An important next stepwould be to bring our metrics to these broader notions of invariance and equivariance.

Original: Future research should explore how invariance to more complex sets of transformationscan be measured without losing too much of the associated rigor in the metrics. Despite our focus on classiﬁcation models in computer vision in this paper, there is no reason that 
Revised: It would beparticularly valuable if our metrics could be modiﬁed so that they can measure equivariance withoutexplicitly specifying a group action in a hidden layer.

Original: G -EED cannot be applied to other model types in other domains. Testing G -EED in other MLsettings where the concepts of invariance or equivariance are important would bolster the evidencethat G -EED is an important evaluation tool to use when studying deep neural networks.  Conclusion
Revised: This would enable us to measure G -EED foremergent types of learned equivariance where the action of G on the output space is not speciﬁed bythe user.

Original: In this paper we described a novel family of metrics meant to measure equivariance in deep learningmodels.
Revised: In this paper we described a novel family of metrics meant to measure equivariance in deep learningmodels.

Original: We use these metrics to suggest answers to questions related to the extent to which neuralnetworks are or are not invariant or equivariant to different types of transformations.
Revised: We use these metrics to suggest answers to questions related to the extent to which neuralnetworks are or are not invariant or equivariant to different types of transformations.

Original: Surprisingly,while invariance and equivariance are fundamental topics within machine learning, this is one of theﬁrst works (to our knowledge) that tries to measure these properties empirically in modern neuralnetworks.
Revised: Surprisingly,while invariance and equivariance are fundamental topics within machine learning, this is one of theﬁrst works (to our knowledge) that tries to measure these properties empirically in modern neuralnetworks.

Original: We hope that this work will open a conversation on this important topic that will ultimatelylead to a better understanding of how deep learning is able to achieve the results that it does.
Revised: We hope that this work will open a conversation on this important topic that will ultimatelylead to a better understanding of how deep learning is able to achieve the results that it does.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’scontributions and scope?
Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’scontributions and scope?

Original: [Yes](b) Did you describe the limitations of your work?
Revised: [Yes](b) Did you describe the limitations of your work?

Original: [Yes] See Section 6(c)
Revised: [Yes] See Section 5(c)

Original: Did you discuss any potential negative societal impacts of your work?
Revised: Did you discuss any potential negative societal impacts of your work?

Original: [No] As thework investigates how deep learning algorithms work in a non-application speciﬁcsetting, we did not see this as relevant.(d)
Revised: [No] As thework investigates how deep learning algorithms work in a non-application speciﬁcsetting, we did not see this as relevant.(d)

Original: Have you read the ethics review guidelines and ensured that your paper conforms tothem?
Revised: Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]2.
Revised: [Yes]2.

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: In both
Revised: In both

Original: Proposition 3.1 and Proposition 3.2 we give a rigorous and speciﬁc set of assumptions.
Revised: Proposition 3.1 and Proposition 3.2 we give a rigorous and speciﬁc set of assumptions.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] See Appendix A.3and A.
Revised: [Yes] See Appendix A.6and A.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main exper-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main exper-

Original: imental results (either in the supplemental material or as a URL)?
Revised: imental results (either in the supplemental material or as a URL)?

Original: [Yes] We include
Revised: [Yes] We include

Original: explicit deﬁnitions for all metrics in Section 3.1 and include details that would allow the experiments to be recreated in Section 4 and
Revised: explicit deﬁnitions for all metrics in Section 3.1 and include details that would allowthe experiments to be recreated in Section 4 and

Original: Appendix A.
Revised: Appendix A.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e g., data splits, hyperparameters, how theywere chosen)?

Original: [No] We specify all training and dataset details in Appendix A.1 andA.2 except for the training/test splits.
Revised: [No] We specify all training and dataset details in Appendix A.1 andA.2 except for the training/test splits.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] We include conﬁdence intervals in all of our ﬁgures.
Revised: [Yes] We include conﬁdence intervals in all of our ﬁgures.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type
Revised: (d) Did you include the total amount of compute and the type of resources used (e

Original: of GPUs, internal cluster, or cloud provider)?
Revised: g., typeof GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Appendix A.
Revised: [Yes] See Appendix A.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] We cited all datasets
Revised: [Yes] We cited all datasets

Original: and pretrained models that we used in our experiments.
Revised: and pretrained models that we used in our experiments.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] We cite licenses for MNIST, xView,
Revised: [Yes] We cite licenses for MNIST, xView,

Original: CIFAR10, and Omniglot.
Revised: CIFAR10, and Omniglot.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No]
Revised: [No]

Original: The only asset is our version of the public xView dataset, xView Maritime.
Revised: The only asset is our version of the public xView dataset, xView Maritime.

Original: We do
Revised: We do

Original: describe how we created our version of this dataset.
Revised: describe how we created our version of this dataset.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable

Original: [Table]

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: [Table] (a) Did you include the full text of instructions given to participants and screenshots, if[Table]
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: [Table] (b) Did you describe any potential participant risks, with links to Institutional Review[Table]
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: [Table] (c) Did you include the estimated hourly wage paid to participants and the total amount[Table]
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]

Revised: A AppendixA.1 Experimental Details

Revised: All our models were trained on an Nvidia A100 using the Adam optimizer.

Original: [Table]
Revised: Batch size, learning rate(LR), and weight decay hyperparameters are summarized in the table below. Recall from Sections 4.1through 4.3 that we train the CNN and C 8 -CNN (architecture description below) on rotated MNISTand train AlexNet, ResNet50, DenseNet121, and LeViT192 on Maritime xView. Hyperparameterswere chosen based on an informal search. Running our experiments with a more comprehensivehyperparameter search would be a worthwhile future exercise since it might yield insight into howdifferent hyperparameters affect model invariance and equivariance.

Revised: The standard CNN and C 8 -CNN used in Section 4.1 consist of 6 convolutional blocks, each containinga standard convolutional layer, a batch norm, a ReLU nonlinearity, and a pooling layer.

Revised: The C 8 -CNNmodels have an analogous 6 block structure with the difference that each block uses C 8 -rotationequivariant, steerable convolutional layers and a so-called inner batch norm (a batch norm adaptedfor equivariant frameworks [46]).

Revised: Both model types have a ﬁnal linear layer.

Revised: [Table]

Revised: A.2 Dataset Details

Revised: The datasets used in our experiments include (i) rotated MNIST , which consists of images from

Revised: MNIST rotated by random angles [13] (MNIST is covered by a Creative Commons Attribution-Share

Revised: Alike 3.0 license) and (ii) xView maritime , a classiﬁcation dataset we constructed by cropping thebounding boxes of all maritime vessel classes found in xView [31], an overhead imaging dataset(xView is covered by an Attribution-Noncommercial-ShareAlike 4.0 International (CC BY-NC-SA4.0) license).

Revised: In our version of rotated MNIST we excluded the class 9 so that we could removethe drop in model accuracy resulting from the similarity of a 9 and an upside-down 6.

Revised: We apply acircular mask to all images before using them as input to a model so that invariance is not broken bythe artifacts introduced by rotating a rectangular image by θ degrees where θ (cid:54) = 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ .

Revised: See the bottom right image in Figure 5 for an example of this.

Revised: Below we describe our OOD datasets for Section 4.2.

Revised: • MNIST Train:

Revised: The original MNIST training images.

Revised: • CIFAR10: Grayscale versions of the CIFAR10 images [28] (CIFAR10 is covered under an

Revised: MIT license ).

Revised: • MNIST (Noisy) 1-6: MNIST images perturbed by noise randomly sampled uniformly from

Revised: intervals (1) [0 , 0 . 2] , (2) [0 , 0 . 4] , (3) [0 , 0 . 6] , (4) [0 , 0 . 8] , (5) [0 , 1 . 0] , and (6) [0 , 1 . 2] .

Revised: Notice

Revised: that any of these noise perturbations can push pixels outside of image bounds.

Revised: We generally

Revised: assume that images with more noise are farther OOD.

Revised: • MNIST (blurry): MNIST images with Gaussian blur applied.

Revised: • Omniglot: Another image classiﬁcation dataset that contains characters [30] (Omniglot is

Revised: covered under an MIT license).

Revised: • Inverted MNIST: MNIST where each pixel has had the transformation f ( x ) = 1 − x

Revised: applied to it.

Revised: This inverts the intensity of pixels to make the MNIST

Revised: dataset look more like

Revised: Omniglot (the background has high intensity and the digits have low intensity).

Revised: A.3 How does invariance in a model’s latent space change when augmentation is or is notused?

Revised: As in Section 4.3, we focus on the case where G = C 8 , the group generated by 45 ◦ rotations, andthe dataset is xView Maritime.

Revised: We trained 5 ResNet50s, 5 AlexNets, and 5 DenseNet121s with andwithout rotation augmentation respectively.

Revised: We then measured latent space C 8 -EED in all models.

Revised: Our results can be found in Figure 11.

Revised: We ﬁnd that models trained without augmentation indeed hadconsistently lower C 8 -EED both when we used our standard Euclidean distance metric and when weused the cosine similarity version.

Revised: A.4 The invariance of supervised and self-supervised models

Revised: In an additional experiment, we compared the invariance of supervised models to the invarianceof self-supervised models.

Revised: To do this we evaluated four different convolutional neural networksin one of the simplest possible settings where G is the order 2 group of reﬂections across thevertical axis.

Revised: The models we chose to test were a standard ResNet50 (we used the Torchvisionpretrained weights)

Revised: [20; 35], a Big Transfer ResNetV2 [26] (with weights from ‘PyTorch Image

Revised: Models’ [49]), a ResNet50 trained using the self-supervised method DINO (we used the author’spretrained weights) [3], and a ResNet50 trained using the self-supervised method SimCLR [6] (weused weights from [45]).

Revised: Note that the ﬁrst two models utilized supervised training and the secondtwo utilized self-supervised training techniques.

Revised: Further, SimCLR is trained with a contrastive lossthat directly optimizes for invariance of input to speciﬁc transformations.

Revised: On the other hand, DINOuses ‘knowledge distillation’, without a contrastive loss.

Revised: We choose to concentrate on latent space

Revised: G -EED in this experiment.

Revised: Our results can be found in Figure 12.

Revised: Note that since all of these models have a 2048 -dimensionallatent space, we can directly compare them.

Revised: This is something we could not do in Section 4.3.

Revised: We see that in terms of both versions of latent space G -EED (that which uses Euclidean distanceand that which uses cosine similarity), the model trained using SimCLR has signiﬁcantly moreinvariance.

Revised: This is not surprising given that the contrastive loss that SimCLR uses explicitly optimizesfor similarity between an image and its reﬂection.

Revised: DINO, which is also trained in a self-supervisedmanner, but which does not explicitly use the contrastive loss is less invariant (how it compares tothe two supervised models depends on which metric is used).

Revised: This suggests that use of contrastiveloss can boost the invariance properties of a model but that self-supervision alone may not alwaysyield invariance properties different from supervised training.

Revised: This is an area that could beneﬁt fromfurther, more comprehensive investigation.

Revised: A.5 Does learned equivariance involve re-ordering of tensor channels?

Revised: Suppose that the function f (cid:96) : X → R C (cid:96) × H (cid:96) × W (cid:96) corresponds to the ﬁrst (cid:96) layers of a network,terminating with a 3 -tensor which has C (cid:96) channels of height H (cid:96) and width W (cid:96) .

Revised: Suppose that G isa group that acts both on the input space X and the hidden space R C (cid:96) × H (cid:96) × W (cid:96) .

Revised: The channelwise

Revised: G -EED metric that we proposed in this paper, assumes that when we compare channels from f (cid:96) ( gx )with channels gf (cid:96) ( x ) , we should assume the trivial bijection.

Revised: That is, we should compare the ﬁrstchannel of f (cid:96) ( gx ) , [ f (cid:96) ( gx )] 1 , with the ﬁrst channel of gf (cid:96) ( x ) , g [ f (cid:96) ( x )] 1 , the second channel [ f (cid:96) ( gx )]

Revised: 2with the second channel g [ f (cid:96) ( x )] 2 , etc.

Revised: On the other hand, in many equivariant CNN’s, the group action of g on f (cid:96) ( x ) not only changesindividual channels, it also permutes their order.

Revised: It is reasonable to ask whether CNNs trained withaugmentation might learn some similar “emergent” structure not only within individual channels, butalso among them.

Revised: The channelwise G -EED metric would likely not detect this kind of equivariance.

Revised: In this section we examine this possibility.

Revised: Though we do not disprove its existence, we run severalpreliminary experiments that suggest that this is not likely.

Revised: This is an area that would beneﬁt fromadditional study.

Revised: In our preliminary investigation, we focus on the ﬁlters at each layer (rather than input and output).

Revised: We chose to do this based off of the observation that in equivariant CNNs that utilize the regular representation and which are designed to display the phenomenon we are looking for, ﬁlters come inentire orbits.

Revised: That is, if w is a ﬁlter in a layer, then so is each element in the orbit Gw (i.e., the orbitof w under the action of G ).

Revised: In Figure 13 for example, four ﬁlters of a C 4 -equivariant convolutionallayer [46] are displayed.

Revised: The ﬁlters represent an orbit under the action of all 90 -degree rotations (e.g.,

Revised: C 4 ).

Revised: As a ﬁrst step towards identifying structured equivariance in augmentation trained CNNs, we compared all channels of each pair of ﬁlters at a given convolutional layer of an AlexNet CNN.

Revised: However,comparing all the possible pairings is combinatorially prohibitive, even for simple architectures like

Revised: AlexNet.

Revised: In Figure 14, we plot the metric

Revised: [Equation]

Revised: 7) where w (cid:96),i,t is the height H (cid:48) (cid:96) and weight W (cid:48) (cid:96) 2 -tensor obtained by taking channel t from ﬁlter i inlayer (cid:96) and g ( w (cid:96),i,t ) is the action of group element g on w (cid:96),i,t .

Revised: It this case simply rotation by amultiple of 90 ◦ .

Revised: We report the average of this metric over all convolutional layers of four different

Revised: AlexNet models trained on MNIST: AlexNet trained from scratch without rotation augmentation,

Revised: AlexNet trained rotation augmentation but no pretraining, AlexNet with both pretraining and rotationaugmentation, and ﬁnally Alex with C 8 -equivariant layers [46].

Revised: Random i and t are selected foreach computation, and for computational simplicity we only evaluated on the subgroup C 4 of C 8generated by 90 -degree rotations.

Revised: A network where some ﬁlters are rotations of others would be expected to achieve a value of 0 for (7).

Revised: Indeed, we can see that this is what happens to the C 8 -equivariant network.

Revised: We see that the othernetworks do not achieve zero.

Revised: Indeed, with the exception of the pretrained AlexNet, for which (7)increases slightly over training, the other non-equivariant networks do not change signiﬁcantly at all.

Revised: This indicates that this form of emergent equivariance does not emerge in this example.

Revised: A.6 A Proof of Proposition 3.

Revised: Proof.

Original: {+To prove this, consider the function f : R 2 → R deﬁned such that for ( x, y ) ∈ R 
+}
Revised: {+To prove this, consider the function f : R 2 → R deﬁned such that for ( x, y ) ∈ R 
+}

Revised: [Equation]

Revised: The order 2 cyclic group Z 2 = { 1 , σ } acts on R 2 by permuting coordinates.

Revised: That is, theonly nonidentity element σ acts on R 2 by sending ( x, y )

Revised: (cid:55)→ ( y, x ) .

Revised: It is clear that f is

Revised: Z 2 -invariant.

Revised: However, note that if f 1 : R 2 → R 2 is deﬁned by f 1 ( x, y )

Revised: =

Revised: (2 x, y ) and f 2 : R 2 → R isdeﬁned by

Revised: [Equation]

Revised: then f = f 2 ◦ f 1 .

Revised: But neither f 1 nor f 2 is Z 2 -equivariant to the actions described above.2.

Revised: This proof follows easily from the deﬁnitions.

Revised: Suppose that f 1 is G -invariant, then for anyx ∈ X and g ∈ G , f 1 ( gx ) = f 1 ( x ) .

Revised: Hence

Revised: [Equation]

Revised: A.7 A Proof of Proposition 3.2

Revised: Proof.

Revised: First note that for ﬁxed x and g , m ( f ( gx ) , g ˆ f ( x ))

Original: {+= 
+}
Revised: {+= 
+}

Revised: if and only if f ( gx ) =

Revised: gf ( x ) .

Revised: This follows from the fact that since G acts faithfully on Y , ˆ f ( x ) =

Revised: f

Revised: ( x ) .

Revised: First assume thatm ( f ( gx ) , g ˆ f ( x ))

Revised: = 0 .

Revised: Then, since m is a metric, f ( gx ) =

Revised: g ˆ f ( x ) =

Revised: gf ( x ) giving the desired result.

Revised: Next, if f ( gx ) =

Revised: gf ( x ) =

Revised: g ˆ f ( x ) , then it again follows that m ( f ( gx ) , g ˆ f ( x ))

Revised: = 0 .

Revised: Next we recall the basic fact [16, proposition 2.16] from measure theory, which states that ifh : W → [0 , ∞ ] is a non-negative measurable function on measure space ( W, ν ) , then

Revised: [Equation]

Revised: almost everywhere.

Revised: Recall that

Revised: [Equation]

Revised: where φ Y is the representation of G on Y and µ ker( φ Y ) is the normalized Haar measure on thesubgroup ker( φ Y )

Revised: ⊂ G .

Revised: Let W =

Revised: X

Revised: × G × ker( φ Y ) with ξ = ν

Revised: × µ × µ ker( φ Y ) the product measureon W generated by Haar measure µ on G , µ ker( φ Y ) , and the probability measure ν associated with D .

Revised: Note that f is measurable by virtue of being continuous, G acts linearly and hence is measurable, andm is measurable by virtue of being a metric.

Revised: It follows that the function h : X × G × G → [0 , ∞ ] deﬁned by

Revised: [Equation]

Revised: is measurable.

Revised: The result then follows from the observation that m is non-negative (and hence h is)and from (8).

Revised: That is, if (3) is zero, then since m ( f ( gx ) , g ˆ f ( x )) is non-negative, then (8) tells us thatm ( f ( gx ) , g ˆ f ( x ))) is zero almost-everywhere with respect to measure ξ .

Revised: On the other hand, (8) alsosays that if m ( f ( gx ) , g ˆ f ( x )) is zero almost everywhere (including the case where this term is zeroeverywhere), then (3) is equal to zero.

Revised: A.8 The Reason for Normalization of Latent Space G -EED

Revised: We normalize E latent ( f, G ) by M because when we use (cid:96) 2 -distance to compute G -EED withoutnormalization, it is sensitive to scaling in a way that is unrelated to downstream task performance.

Revised: Toillustrate, note that by scaling f 1 by c > 1 , the unnormalized latent space G -EED increases, indicatinga decrease in G -invariance.

Revised: More precisely, if f (cid:48) 1 = cf 1 , then E ( f (cid:48) 1 , G )

Revised: =

Revised: c E ( f 1 , G ) .

Revised: However,if we set f (cid:48) 2 = 1 c f 2 , then f (cid:48) 2 ◦ f (cid:48) 1 = f = f 2 ◦ f 1 .

Revised: Thus, f 1 can be made to have arbitrarily largeunnormalized latent space G -EED while the total model f remains constant.

Revised: A different but equallyillustrative example is visualized in Figure 15.

Revised: Two feature extractors have identical unnormalizedlatent space C 8 -EED for a single rotation orbit of an image of a 4 (average distance between bluepoints and their centroid), but the second feature extractor closely clusters points in the orbit relativeto other instances from MNIST, while the ﬁrst feature extractor does not.

Revised: We would argue that inmost cases the second feature extractor should be called more invariant with respect to this particulartask.

Revised: Normalization mitigates this issue, giving a more reasonable notion of latent space invariance inthe context of machine learning.

Revised: A.9 Channelwise C 8 -EED for out-of-distribution datasets

Revised: In this section we provide Figure 16, which shows the channelwise C 8 -EED for collections of: CNNmodels with random weights, CNN models that have been trained on MNIST, and C 8 -CNN modelsthat have been trained on MNIST.

Revised: In Figure 16 we show the channelwise C 8 -EED at layers 2 and 5 for each of the model types.

Revised: Unsurprisingly, the C 8 -CNN has high C 8 -equivariance for all layers, whereas both the trained anduntrained CNNs have substantially lower C 8 -equivariance.

Revised: Across layers, the channelwise C 8 -EEDremains fairly constant across datasets for the untrained models.

Revised: However, the channelwise C 8 -EED for early layers of the trained CNNs differs across datasets (with equivariance on OOD datasetsgenerally being less than the equivariance on the training set MNIST).

Revised: In later layers, however, thedifference in C 8 -equivariance between MNIST and an OOD dataset is negligible (Figure 16, right).

Revised: This suggests that CNNs do learn some minimal amount of equivariance at early layers, but thisequivariance either dissipates or becomes undetectable at later layers.