Title: Learning symmetries via weight-sharing with doubly stochastic tensors
Venue: NeurIPS 2024
Accepted: yes
Keywords: symmetry discovery

weight-sharing
Abstract: Group equivariance has emerged as a valuable inductive bias in deep learning, enhancing generalization, data efficiency, and robustness. Classically, group equivariant methods require the groups of interest to be known beforehand, which may not be realistic for real-world data. Additionally, baking in fixed group equivariance may impose overly restrictive constraints on model architecture. This highlights the need for methods that can dynamically discover and apply symmetries as soft constraints. For neural network architectures, equivariance is commonly achieved through group transformations of a canonical weight tensor, resulting in weight sharing over a given group $G$. In this work, we propose to *learn* such a weight-sharing scheme by defining a collection of learnable doubly stochastic matrices that act as soft permutation matrices on canonical weight tensors, which can take regular group representations as a special case. This yields learnable kernel transformations that are jointly optimized with downstream tasks. We show that when the dataset exhibits strong symmetries, the permutation matrices will converge to regular group representations and our weight-sharing networks effectively become regular group convolutions. Additionally, the flexibility of the method enables it to effectively pick up on partial symmetries.

Learning symmetries via weight-sharing with doubly stochastic tensors

Abstract

Group equivariance has emerged as a valuable inductive bias in deep learning, enhancing generalization, data efficiency, and robustness. Classically, group equivariant methods require the groups of interest to be known beforehand, which may not be realistic for real-world data. Additionally, baking in fixed group equivariance may impose overly restrictive constraints on model architecture. This highlights the need for methods that can dynamically discover and apply symmetries as soft constraints. For neural network architectures, equivariance is commonly achieved through group transformations of a canonical weight tensor, resulting in weight sharing over a given group G . In this work, we propose to learn such a weightsharing scheme by defining a collection of learnable doubly stochastic matrices that act as soft permutation matrices on canonical weight tensors, which can take regular group representations as a special case. This yields learnable kernel transformations that are jointly optimized with downstream tasks. We show that when the dataset exhibits strong symmetries, the permutation matrices will converge to regular group representations and our weight-sharing networks effectively become regular group convolutions. Additionally, the flexibility of the method enables it to effectively pick up on partial symmetries.

1 Introduction

Equivariance has emerged as a beneficial inductive bias in deep learning, enhancing performance across a variety of tasks. By constraining the function space to adhere to specific symmetries, models not only generalize better but also achieve greater parameter efficiency [8, 9]. For instance, integrating group symmetry principles into generative models has enhanced sample generation and efficient learning of data distributions, particularly benefiting areas such as vision and molecular generation [12, 5].

The most well-known and transformative models in equivariant deep learning are convolutional neural networks (CNNs) [18], which achieve translation equivariance by translating learnable kernels to every position in the input. This design ensures that the weights defining the kernels are shared across all translations, so that if the input is translated, the output features are correspondingly translated; in other words, equivariance is achieved through weight sharing . In the seminal work by Cohen and Welling [9], this concept was extended to generalize weight-sharing under any discrete group of symmetries, resulting in the group-equivariant CNN (G-CNN). G-CNNs enable G-equivariance to a broader range of symmetries, such as rotation, reflection, and scale [9, 4, 35, 29], thereby expanding the applicability of CNNs to more complex data transformations.

However, the impact of G-CNNs is closely tied to the presence of specific inductive biases in the data. When exact symmetries, such as E(3) group symmetries in molecular point cloud data, are known to exist, G-CNNs excel. Yet, for many types of data, including natural images and sequence data, these exact symmetries are not present, leading to overly constrained models that can suffer in performance [32, 24, 2, 31]. In scenarios with limited data and for certain critical downstream tasks, having appropriate inductive biases becomes even more crucial.

To avoid overly constraining models, symmetries must be chosen carefully to match those in the input data. This requires prior knowledge of these symmetries, which may not always be available. Furthermore, different symmetries at different scales can coexist, making manual determination of these symmetries highly impractical. To address this, multiple works have proposed partial or relaxed G-CNNs [24, 6, 2, 32]. Such models are initialized to be fully equivariant to some groups and learn from the data to partially break equivariance on a per-layer basis where necessary. However, these methods still require specifying which symmetries to include and can only achieve equivariance to subsets of these symmetries.

In this work, we tackle the challenge of specifying group symmetries upfront by introducing a general weight-sharing scheme. Our method can represent G-CNNs as a special case but is not limited to exact equivariance constraints, offering greater flexibility in handling various symmetries in the data. Inspired by the idea that group equivariance for finite groups can be achieved through weight-sharing patterns on a set of base weights [23], we propose learning the symmetries directly from the data on a per-layer basis, requiring no prior knowledge of the possible symmetries.

We leverage the fact that regular group representations act as permutations and that the expectation of random variables defined over this set of permutations is a doubly stochastic matrix [7]. This implies that regular partial group transformation can be approximated by a stack of doubly stochastic matrices which essentially act as (soft) permutation matrices. Consequently, we learn a set of doubly stochastic matrices through the Sinkhorn operator [28], resulting in weight-sharing under learnable group transformations.

We summarize our contributions as follows:

We propose a novel weight-sharing scheme that can adapt to group actions when certain symmetry transformations are present in the data, enhancing model flexibility and performance.

Wepresent empirical results on image benchmarks, demonstrating the effectiveness of our approach in learning relevant weight-sharing schemes when there are clear symmetries.

The proposed method outpaces models configured with known symmetries in environments where they are only partially present. Moreover, in the absence of predefined symmetries, it adeptly identifies effective weight-sharing patterns, matching the performance of fully flexible, non-weight-sharing models.

We provide analyses of the learned symmetries on some controlled toy settings.

2 Related work

Partial or relaxed equivariance Methods such as [24, 6, 2] learn partial equivariance by learning distributions over transformations, and thereby aim to learn partial or relaxed equivariances from data by sampling some group elements more often than others. [32, 33] relax equivariance by introducing learnable equivariance-breaking components. [11, 31] Relax equivariance constraints by parameterizing layers as (linear) combinations of fully flexible, non-constrained components and constrained equivariant components. Finally, several works model soft invariances through learning the amount of data augmentation in the data or model relevant for a given task, either through learned distributions on the group or (automatic) hyperparameter selection [6, 13, 22]. However, these methods require pre-specified sets of symmetry transformations and/or group structure to be known beforehand. In contrast, we aim to pick up the relevant symmetry transformations during training.

Symmetry discovery methods Several methods have been developed for symmetry discovery by leveraging the infinitesimal generators of symmetry groups, as articulated through the Lie algebra [10, 36, 37, 21]. Works such as [10, 36, 21] focus on discovering symmetries associated with

general linear actions. In contrast, [37] extends these concepts to encompass non-linear symmetry transformations, offering capabilities for discovering partial symmetries within the data.

[26, 20] Learn the group structure via (irreducible) group representations. [26] Proposed to learn the Fourier transform of finite compact commutative groups and their corresponding bispectrum by learning to separate orbits on our dataset. This approach can be extended to non-commutative finite groups leveraging advanced unitary representation theory [20]. However, these methods are constrained to finite-dimensional groups and require specific orbit-predicting datasets. In contrast, our approach learns a relaxation of regular group representations -as opposed to irreducible representations. Moreover, our approach is not merely capable of learning symmetries, it subsequently utilizes them in a regular group-convolution-type architecture.

Weight-sharing methods Previous studies have demonstrated that equivariance to finite groups can be achieved through weight-sharing schemes applied to model parameters. Notably, the works in [23], [39], and [38] provide foundational insights into this approach. In [39], weight-sharing patterns are learned by using a matrix that operates on flattened canonical weight tensors, effectively inducing weight sharing. They additionally prove that for finite groups, there are weight-sharing matrices capable of implementing the corresponding group convolution. However, their approach requires learning these patterns through meta-learning and modeling the weight-sharing matrix as an unconstrained tensor. In contrast, our method learns weight sharing directly in conjunction with the downstream task and enforces the matrix to be doubly stochastic, thereby representing soft permutations by design.

[38] Presents an approach closely aligned with ours, where a weight-sharing scheme is learned that is characterized by row-stochastic entries. Their method involves both inner- and outer-loop optimization and demonstrates the ability to uncover relevant weight-sharing patterns in straightforward scenarios. However, their approach does not support joint optimization of the canonical weights and weight-sharing pattern, and they acknowledge difficulties in extending their method to higher input dimensionalities. Unlike [38], we enforce both row and column stochasticity. Additionally, we can optimize for the sharing pattern and weight tensors jointly, and successfully apply our approach to more interesting data domains such as image processing.

In [30], group actions are integrated directly into the learning process of the downstream task. This method involves learning a set of generator matrices that operate via matrix multiplication on flattened input vectors. However, this approach constrains the operators to members of finite cyclic groups, which inherently limits their ability to represent more complex group structures. Furthermore, this restriction precludes the possibility of modeling partial equivariances, reducing the flexibility and applicability of the model to more diverse or complex scenarios.

3 Background

We begin by revisiting group convolutional methods in the context of image processing, followed by their relation to weight-sharing schemes. We then proceed to briefly cover the Sinkhorn operator, which is the main mechanism through which we acquire weight-sharing schemes. Some familiarity with group theory is assumed, and essential concepts will be outlined in the following.

Groups We are interested in (symmetry) groups, which are algebraic constructs that consist of a set G and a group product-which we denote as a juxtaposition-that satisfies certain axioms, such as the existence of an identity element e ∈ G such that for all g ∈ G we have eg = ge = g , closure such that for all g, h ∈ G we have gh ∈ G , the existence of an inverse g -1 for each g such that g -1 g = e , and associativity such that for all g, h, i ∈ G we have ( gh ) i = g ( hi ) .

Representations In the context of geometric deep learning [8], it is most useful to think of groups as transformation groups, and the group structure describes how transformations relate to each other. Specifically, group representations ρ : G → GL ( V ) are concrete operators that transform elements in a vector space V in a way that adheres to the group structure (they are group homomorphisms). That is, to each group element g , we can associate a linear transformation ρ ( g ) ∈ GL ( V ) , with GL ( V ) the set of linear invertible transformations on vector space V .

Group convolution Concretely, such representations can be used to define group convolutions. Consider feature maps f : X → R D over some domain on which a group action is defined, i.e., over a G -space . E.g., for images (signals over X = R 2 ) we could consider the group G = ( R 2 , +) of translations, which acts on X via gx = x + y , with g = ( y ) a translation by y ∈ R 2 . While the group G merely defines how two transformations g, h ∈ G applied one after the other correspond to a net translation gh ∈ G , a representation ρ concretely describes how data is transformed. E.g., signals f : X → R can be transformed via the left-regular representation [ ρ ( g ) f ]( x ) := f ( g -1 x ) , which in the case of images and the translation group is given by [ ρ ( g ) f ]( x ) = f ( x -y ) .

In general, group convolution is defined as transforming a base kernel under every possible group action, and for every transformation taking the inner product with the underlying data via

G

with ⟨· , ·⟩ denoting the inner product. For images, which are essentially functions over the group G = ( R 2 , +) , and taking ⟨ k, f ⟩ := ∫ X k ( x ) f ( x )d x the standard inner product, Eq. (1) boils down to the standard cross-correlation operator 2 :

G

Standard ( R , +)

Semi-direct product groups When equivariance to larger symmetry groups is desired, e.g. in the case of G = SE (2) roto-translation equivariance for images with domain X = R 2 , a lifting convolution can be used to generate signals over the group G . In essence it is still of the form of (2), however integration is over X instead of over G :

SE (2)

with g = ( x, R ) ∈ ( R 2 , +) ⋊ SO (2) . The roto-translation group is an instance of a semi-direct product group (denoted with ⋊ ) between the translation and rotation group, which has the practical benefit that a stack of rotated kernels can be precomputed [9], and the translation part efficiently be taken care of via optimized Conv2D operators. Namely via ( k ⋆ f )( x, R i ) = Conv2D [ k i , f ] , with k i := k ( R -1 i x ) . This trick can also be applied for full group convolutions (2).

4 Method

Our objective is to uncover the underlying symmetries of datasets whose exact symmetries may not be known, in a manner that is both parameter-efficient and free from rigid group constraints. To achieve this, we re-examine regular representations and their critical role in generating various instantiations of the fundamental group convolution equation (1). Moving from the continuous setting to concrete instantiations, we derive weight-sharing from a finite-dimensional vector of base weights by interpreting regular representations as permutations. We further analyze the characteristics of this weight-sharing approach, proposing the use of doubly stochastic matrices. This analysis forms the foundation for developing weight-sharing layers that adaptively learn dataset symmetries.

4.1 Weight-sharing through permutations

In the current section, we first establish the connection between representations, weight-sharing and permutations. We then proceed to provide practical instantiations as ingredients for the proposed weight-sharing convolutional layers.

Weight-sharing through learnable representations To achieve weight-sharing over a finite set of symmetries, we define learnable representations ρ : G → GL ( V ) . Specifically, we assign a learnable transformation (permutation matrix) to each element in G . It is important to note that we refer to G and ρ as a "group" and "representation" in a loose sense, as we relax the homomorphism property and do not initially endow G with a group product. Consequently, the collection of linear transformations does not form a group representation a priori. However, our proposed method is capable of modeling this structure in principle.

Regular representations as permutations Consider the case of a continuous linear Lie group G , e.g. of rotations SO (2) , and a real signal f : G → R over it. This signal is to be considered an infinite dimensional vector with continuous "indices" g ∈ G that index the vector elements f ( g ) . To emphasize the resemblance of the regular representation to permutation matrices we write it as

with P g ( i, h ) = δ g -1 i ( h ) a kernel that for each g maps h to a new "index" i ∈ G , where the Dirac delta essentially codes for the group product as δ g -1 h ( i ) is non-zero only if g -1 i = h ⇔ g · h = i .

The discrete counterpart of such an integral transform is matrix-vector multiplication with a matrix ρ ( g ) = P g with entries P ghi = 1 if gh = i and zero otherwise. As a concrete example, consider a signal f : G → R over the discrete group G = C 4 of cyclic permutations of size 4, i.e., a periodic signal of length four, then we could vectorize it as f → v with entries v g = f ( g ) and the regular representation becomes a simple cyclic permutation matrix with

We refer to the collection of permutation matrices P ∈ [0 , 1] | G |×| G |×| G | as the permutation tensor .

Recall that the regular convolution operator (2) is not only defined for signals over groups G but for G -spaces X , in general, as in Eq. (4). It requires a representation that acts on signals (convolution kernels) over X . However, since the group G acts by automorphisms (bijections) on the space X , we can define the regular representation-as before-using permutation integral with P g ( x, x ′ ) = δ g -1 x ′ that effectively sends old "indices" x ′ to their new location x , or concretely via permutation matrices P of shape | G | × | X | × | X | , where X is the domain of the signal that is transformed-which can also be G .

In practice, it is common that we perform a discretization of continuous signals f , e.g., when we work with images, we discretize the continuous signal f : R 2 → R C to f : Z 2 → R C . Therefore, the group action over the discretized signal should approximate the group action over the continuous signal. We can see that in some cases, the group representation of the action on the discrete signal can still be implemented using permutation matrices. E.g., 90 ◦ rotations (in C 4 ) applied to images merely permute the pixels. However, for finer discretizations, e.g. using 45 ◦ rotations, interpolation can be used as a form of approximate permutations [17, Fig. 2].

Weight sharing In a discrete group setting, we then see that convolution (1) is obtained by multiplications with matrices obtained by stacking permuted base kernel weights w ∈ R | X |

Regular representations allow for element-wise activations An important practical element of using regular representations to define group convolutions is that permutations commute with element-wise activations, namely, [ ρ ( g ) σ ( f )]( i ) = σ ( f )( g -1 i ) = σ ( f ( g -1 i )) = σ ([ ρ ( g ) f ]( i )) . In contrast, steerable methods-based on irreducible representations (cf. App. A.1)-require specialized

R1 · vec(k)

R2 · vec(k)

reshape

R3 · vec(k)

R4 · vec(k)

k:R² ×H

activation functions so as not to break group equivariance. Such activations in practice are not as effective as the classic element-wise activations such as ReLU [34]: they may introduce discretization artifacts that disrupt exact equivariance, ultimately constraining the method's expressivity [5]. Hence, when learning weight-sharing schemes-as is our objective-it is preferred to achieve weight-sharing using regular representations without the risk of breaking equivariance by using standard activation functions.

4.2 Learnable doubly stochastic tensors

Having established the link between regular representations and soft permutations, we now motivate the use of doubly stochastic matrices as natural candidates for their implementation. Specifically, we utilize the fact that the expected value of random variables over this set of permutations yields a doubly stochastic matrix [7].

Doubly stochastic matrices Let S ∞ ( S n ) denote respectively the system of infinite ( n × n ) doubly stochastic matrices, i.e. matrices S ≡ { s ij ∈ [0 , 1] : i, j = 1 , 2 , ... ( , n ) } such that ∑ j s ij = 1 , and ∑ i s ij = 1 . Let P ∞ ( P n ) denote respectively the system of infinite ( n × n ) permutation matrices, i.e. matrices P ≡ { p ij ∈ { 0 , 1 } : i, j = 1 , 2 , ... ( , n ) } such that ∑ j p ij = 1 , and ∑ i p ij = 1 . Note that for any permutation tensor P ∈ R | G |×| X |×| X | , where X is the domain of the signal that is transformed by the group G , then P g ∈ P | X | for every g ∈ G .

Then, by Birkhoff's Theorem [7], and its extension to infinite dimensional matrices, commonly called Birkhoff's Problem 111 [14, 25, 15, 3], we have that any convex combination of permutation matrices will be equal to a doubly stochastic matrix, i.e,

where λ ( P ) gives a probability measure supported on a finite subset of the set of permutation matrices P . Therefore we may state that

Using doubly stochastic matrices, we can model approximate equivariance as defined in [24].

I.e., let S be a random variable over { P g ∈ P | X | | g ∈ G } with a finitely supported probability measure P [ S = P g ] = λ ( P g ) for every g ∈ G , then S = E [ S ] = ∑ g ∈ G P [ S = P g ] P g is a doubly stochastic matrix. We want to note that S can be seen as a generalization of the convolution matrix presented in [19].

Sinkhorn operator The Sinkhorn operator [28, 1] transforms an arbitrary matrix to a doubly stochastic one through iterative row and column normalization, provided that the number of iterations

R1

R2

R3

R4

C1

C8

90

180

270

is large enough. That is, initialize a tensor X ∈ R N × N , then it will converge to a doubly stochastic tensor via the following algorithm:

with T c and T r the normalization operators over the rows and columns, respectively, defined as T c = X ⊘ 1 N 1 T N X ︸ ︷︷ ︸ sum c ( X ) and T r = X ⊘ X1 N 1 T N ︸ ︷︷ ︸ sum r ( X ) , where ⊘ denotes elementwise division, sum c ( · ) , sum r ( · ) perform column-wise and row-wise summation, respectively.

Our proposal: Weight Sharing Convolutional Neural Networks Having established the foundational elements, we now define our Weight Sharing Convolutional Neural Networks (WSCNNs). We let Θ l i ∈ R | X |×| X | be a collection of learnable parameters that parametrize the representation stack of the layer l as R l = ( S K (Θ l 0 ) , S K (Θ l 1 ) , ..., S K (Θ l N )) T ∈ [0 , 1] | G |×| X |×| X | . i.e., we parameterize this tensor as a stack of | G | approximate doubly stochastic | X | -dimensional matrices, wherein stochasticity is enforced via K applications of the Sinkhorn operator. We also define a set of learnable base weights θ l ∈ R | X |× C out × C in . The WSCNN layer is then simply given by (7) with P and w respectively replaced by R l and θ l .

We further note that on image data | X | can be large, making the discrete matrix form implementation computationally demanding. Hence, we consider semi-direct product group parametrizations for G , in which we let G be of the form ( R n , +) ⋊ H , with H a learnable (approximate) group. Then, the representation stacks will merely be of shape | H | × | X ′ | × | X ′ | , with | H | ≪| G | the size of the sub-group and | X ′ | the number of pixels that support the convolution kernel. A WSCNN layer is then efficiently implemented via a Conv2D [ f , R l θ l ] . For group convolutions (after the lifting layer) the representation stacks will be of shape | H | × ( | X ′ | × | H | ) × ( | X ′ | × | H | ) . Computational scaling requirements can be found in Appendix C.4.

5 Experiments

We first demonstrate that the proposed weight sharing method can effectively pick up on useful weight sharing patterns when trained on image datasets with different equivariance priors. We then proceed to show the method can effectively handle settings where partial symmetries are present in the data, and further analyze the learned weight-sharing structures on a suite of toy datasets. Model architectures, regularizers ( norm , ent ) and design choices can be found in Appendix C.4 and C.1, respectively. An analysis of computational requirements can be found in Appendix C.4

5.1 Image datasets and equivariance priors

Weassess the efficacy of our proposed weight-sharing scheme in recognizing data symmetries through experiments on datasets subjected to various data augmentations. Specifically, we evaluate our model on MNIST images that have been rotated (with full SO (2) rotations) and scaled (with scaling

factors between [0 . 3 , 1 . 0] ). We categorize these datasets based on their data symmetry characteristics: MNIST with rotation and scaling as datasets with known symmetries, and CIFAR-10 with flips as a dataset with unknown symmetries.

For RotatedMNIST, we regard a C4-group convolutional model as a benchmark since it has been equipped with a subgroup of the underlying data symmetries a priori. Additionally, we contrast our results with a non-constrained CNN model which has double the number of channels, allowing for free optimization without symmetry constraints. As such, our evaluations are benchmarked against two distinct models: 1) a group convolutional model that is equivariant to discrete rotations, embodying fixed equivariance constraints, and 2) a standard CNN that adheres only to translational equivariance, without additional constraints.

This experimental setup positions the standard CNN as the most flexible model lacking predefined inductive biases. In contrast, the group convolutional neural network (GCNN) is characterized by fixed weight sharing, while our proposed weight-sharing CNN (WSCNN) introduces a semi-flexible, learnable weight-sharing mechanism. Note that we explicitly distinguish between the number of free model and kernel parameters and the additional parameters introduced by our weight sharing scheme throughout results (marked by + ).

Results can be found in Tab. 1. When there is a clear misalignment between the model and data symmetries, the constraints imposed by the model hinder performance, as demonstrated by the C4-GCNN on the scaled MNIST dataset. Notably, our proposed method consistently achieves high performance across all datasets without requiring fixed group specifications. Furthermore, visual inspection of the learned kernels indicates that WSCNN adapts to the underlying data symmetries by effectively rotating kernels, as shown in Figure 2. Additionally, analysis of the learned representation stack reveals that it closely resembles elements of C 4 permutations, further demonstrating the model's capability to internalize and replicate data transformations (see Figure 3 and Appendix B.3).

Additionally, we test the model on CIFAR-10, representing a dataset with possibly more complex/unknown symmetry structures, and CIFAR-10 with horizontal flips (which cannot be represented by C N transformations). Results can be found in Tab. 2, with detailed training and model specifications available in Appendix C.4. We compare against the (possibly misspecified) C4-GCNN, and several unconstrained CNNs models with varying number of channels: 1) CNN-32 matched in free

Model

Rot. Range

GCNN

WSCNN

[0

,

90

◦

]

[0

[0

[0

,

,

,

180

◦

◦

90

180

◦

]

]

]

98

.

84

±

.

002

98

.

98

.

99

.

72

87

25

±

±

±

.

001

.

001

.

001

kernel size, 2) CNN-64 matched in parameter budget, and 3) CNN-128 matched in effective kernel size (calculated as | G | × channels = 4 × 32 = 128 ). Despite the kernel constraints in WSCNN, it achieves performance comparable to that of the unconstrained 128-channel CNN (within < 1% accuracy), at a significantly smaller parameter budget (2.1 M vs. 6.5 M).

5.2 Learning partial equivariances

We show our method is able to pick up on partial symmetries by testing it on MNIST with rotations sampled from a subset of SO(2) and compare it to the C4-GCNN. Additionally, we show results on CIFAR-10 with horizontal flips, which is a commonly used train augmentation. Results can be found in Tab. 3 and Tab. 4.

Additionally, We proceed to test the model's capability to detect data symmetries by applying it to a suite of toy problems, wherein the datasets comprise noisy G -transformed samples. Details on the data generation framework are provided in Appendix B. Our testing employs a single-layer setup aimed at learning a collection of kernels that ideally match each data sample, considering inherent noise. This involves training the model to identify a set of base kernels and their pertinent transformations, effectively adapting to the variations presented by the toy problems.

6 Discussion and Future Work

We demonstrated a method that can effectively identifies underlying symmetries in data, even without strict group constraints. Our approach is uniquely capable of learning both partial and approximate symmetries, which more closely mirrors the complexity found in real-world datasets. Utilizing doubly stochastic matrices to adapt kernel weights for convolutions, our method offers a flexible means of learning representation stacks, accommodating both known and unknown structures within the data. This adaptability makes it possible to detect useful patterns, although these may not always be interpretable in traditional group-theoretic terms due to the absence of predefined structures in the representation stack.

Limitations include computational requirements, which scale quadratically with the size of the group and the kernel size, posing challenges in scenarios with large groups or high-resolution data. As such, in this work we have designed the representation stack to be uniform across the channel dimension. However, this prevents learning of other commonly used image transformations such as color jitter. Furthermore, the need for task-specific regularization to manage entropy scaling during the learning of representations introduces complexity in hyperparameter tuning, which can be a barrier in some applications. Additionally, we observed that representations in later layers may show minimal diversity, suggesting that further innovation in regularization strategies might be necessary to enhance the distinctiveness of learned features across different layers.

For future work, we aim to enhance our method by implementing hierarchical weight-sharing across layers and promoting group equivariance more systematically. One promising direction is to leverage the concept of a Cayley tensor, akin to [20], to identify and reuse learned group structures across different layers of the network. This approach would not only impose a more unified and coherent group structure within the model but also potentially reduce the computational overhead associated with learning separate representations for each layer. By encouraging a shared group structure throughout the network, we anticipate improvements in both performance and interpretability, paving the way for more robust and efficient symmetry-aware learning systems.

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A Preliminaries

A.1 Irreducible representations

In this section, we closely follow the mathematical preliminaries outlined in [34]. For a comprehensive reference on Representation Theory, see [27].

Equivalent representations Two representations ρ and ρ ′ of a group G are considered equivalent if there exists a similarity transform such that:

where Q represents a change of basis matrix.

Irreps A matrix representation is called reducible if it can be decomposed as:

where Q is a change of basis matrix. If the sub-representations ρ 1 and ρ 2 cannot be further decomposed, they are termed irreducible representations ( irreps ). The set of all irreducible representations of a group G is denoted as ˆ G .

Additionally, any representation ρ : G → GL ( V ) of a compact group G can be expressed as:

where I is an index set (possibly with repetitions) over ˆ G .

Similarly to what is showed in Section 4, a representation ρ : G → R d × d can be viewed as a collection of d 2 functions over G . The Peter-Weyl theorem asserts that the collection of functions formed by the matrix entries of all irreps in ˆ G spans the space of all square-integrable functions over G . For most groups, these entries form an orthogonal basis, allowing any function f : G → R to be written as:

where d j is the dimension of the irrep ρ j , while m,n index the entries of ρ j . Note that this expression corresponds to the inverse Fourier transform and that the coefficients w j,m,n can be obtained by the Fourier transform of f with respect to the basis functions { [ ρ j ( g )] mn } j ∈I .

Connection with regular representations It can be shown that the regular representations can be decomposed using the corresponding irreps as follows:

where Q performs the Fourier transform, while Q -1 performs the inverse Fourier transform. This implies that when functions f : G → R are considered as vectors in R | G | , with a basis where each axis corresponds to a group element, then, as we have seen in Section 4, the group action results in a permutation of these axes. However, applying the Fourier transform changes the basis so that G acts independently on different subsets of the axes, resulting in the action being represented by a block-diagonal matrix, which is the direct sum of irreps.

B Toy problems

B.1 Data generation processes

For the construction of the toy problems, we look at two types of data-generating processes:

Equivariant data. Assume a canonical vector ˆ x and corresponding label ˆ y . We assume these vectors transform under a known group G , such that the data-generating process is as follows:

Sample group action:

g ∼ µ ( G ) ,

Apply group action to feature with noise:

x = ρ x ( g ) ˆ x + ϵ

Apply group action to label:

y = ρ y ( g ) ˆ y

with ρ x , ρ y the representations of G acting on the feature and label space, respectively.

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(b) Invariant task. The labels are fixed.

Evaluation metrics To assess whether our method effectively captures the underlying data symmetries, we analyze the learned weight-sharing structure by comparing it to known ground truth patterns. Since our model does not impose associativity or any strict group constraints on the representation stack, it may learn to represent mixtures or interpolations of group elements. Hence, in general, we will not observe a relevant algebraic structure if we produce a Cayley table based on the learned representations as done in [26, 20].

Therefore, we will use an approach that allows us to capture the flexibility of our representations. To this end, we will examine each representation matrix to determine how closely it resembles a convolution matrix [19] associated with a random variable defined over some specific group G . We employ the set of group actions from G , represented as doubly stochastic tensors { P gt k } | G | k =1 , as a reference framework to quantify the fit and alignment of our model's representations with these predefined group actions. As such, for each learned weight sharing tensor R l i , we calculate the fit ˆ P i = ∑ | G | k c k P gt k and acquire coefficients c k > 0 , such that ∑ | G | k c k = 1 in a constrained linear regression setup by minimizing || ˆ P i -R l i || 2 .

B.2 Additional results: toy problems

We conducted experiments on various signals subjected to different transformations, including: a 1D signal with cyclic shifts (exemplary samples shown in Fig. 4a), a 2D signal with C 8 rotations (illustrated in Fig. 7), and a 3D voxel grid enhanced by 24 cube symmetries. In each scenario, the learned kernel stack accurately matched the data samples, achieving perfect accuracy. Figure 6

(a) Shift-dataset with uniformly sampled group elements.

(b) Shift-dataset with partially sampled group elements.

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displays the learned representations for localized shifts in the 1D signal, while Figure 8 presents the learned kernel stack for the 2D signal dataset.

Furthermore, to assess whether our model can identify partial group structures, we evaluate it using two distinct datasets: a 1D signal enhanced with cyclical shifts, and a 3 × 3 × 3 voxel grid subjected to rotations from C 4 × C 4 . For the shift dataset, we utilize the complete set of cyclical shifts as the ground truth representations. Figure 5(a, b) displays the coefficients for the shift dataset. Specifically, part (a) shows coefficients when trained uniformly across group elements, while part (b) illustrates coefficients using only the first half of the group elements, where the group no longer retains cyclic properties or satisfies closure. Given that our method does not assume cyclic groups or group closure, it effectively captures the relevant group transformations even for partial transformations. App. B.4 shows the coefficients for the base representations of C 4 × C 4 × C 4 cube symmetries. Since the data augmentation only applied C 4 × C 4 transformations, the method predominantly identifies elements corresponding to these transformations, as highlighted by the red line.

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B.3 Visualization of G-Conv layers

Figure 9 displays the ground truth permutation matrices that implement a shift-twist operator, which is the group transformation that underlies regular group convolution operator for C 4 rotations. Figure 10 illustrates the corresponding matrices learned for each learnable weight sharing layer on the rotated MNIST dataset. The learned matrices closely show similar patterns as the shift-twist operator, suggesting the model's ability to capture such transformations from training data.

B.4 Additional results: learning partial equivariance

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C Architectural details

In the current section, we outline some design choices used across all experiments. Code is available at https://github.com/computri/learnable-weight-sharing . Firstly, when defining the learnable representation stack R , we have found that anchoring an identity element aids in distinguishing the base kernel from its potential transformations. This approach is based on the intuition of starting with a learnable base kernel and subsequently learning its transformations and we have found that it aids optimization.

C.1 Regularizers

We test two regularizers on the representations R , which are similar to those used by [38]:

Entropy Regularizer : The primary motivation for using the entropy regularizer is to encourage sparsity in our weight-sharing schemes, which helps the matrices approximate actual permutation matrices rather than simply being doubly stochastic. This approach stems from the intuition for some transformations, the weight-sharing schemes should mimic soft-permutation matrices. The effectiveness of this sparsity depends on the specific group transformations relevant to the task-for example, C4 rotations are typically represented by exact permutation matrices. In contrast, C N rotations or scale transformations might require interpolation, thus aligning more closely with soft permutation matrices. Our experimental results indicate that the utility of this regularizer varies with the underlying transformations in the data; for instance, it is not beneficial for scale transformations in the MNIST dataset, as anticipated. The entropy regularizer is of the following form:

Normalization Regularizer : Empirically, we have found the normalization regularizer essential for reducing the number of iterations needed by the Sinkhorn operator to ensure the matrices are row and column-normalized. Without this regularizer, the tensors either fail to achieve double stochasticity or require an excessively high number of Sinkhorn iterations to do so. The normalization regularizer is of the following form:

Where sum r , sum c are defined as in 4.2.

C.2 MNIST

Model architecture For all MNIST experiments, a simple 5-block CNN was used. Each block uses a kernel size of 5 and is succeeded by instance norm and ReLU activation, respectively. After the final convolution block, any spatial and group dimensions are reduced through a global average pooling operation, and a single linear layer is used as a classification head. For the group convolutional model and our weight sharing model, the default hidden channel dimension in the blocks was set to 32 unless otherwise stated, and 64 in the regular CNN models.

Training details The models used a learning rate of 1e-2 and were trained for 100 epochs. All the experiments were done on a single GPU with 24GB memory under six hours.

C.3 CIFAR10

Model architecture We used the ResNet architecture as in [16] Appendix B.1, except that we swapped the final global max pooling operator with a global mean pooling. However, in contrast to [16], we use regular discrete kernels instead of continuous kernel parameterizations.

Training details Following [16], we trained the models for 200 epochs using a learning rate of 1e-4. All the experiments were done on a single GPU with 24GB memory under six hours.

C.4 Computational Demands

Fig. 13 14 show the computational scaling analysis of our weight sharing layer, comparing it to a group convolutional model of the same dimensions. We highlight that regular group convolutions can be implemented via weight-sharing schemes, resulting in equal computational demands for both approaches. Since the weight-sharing approach applies the group transformation in parallel across all elements (as a result of matrix multiplication and reshape operations), our method can prove quite efficient. Regarding memory allocation, group convolutions are often implemented using for-loops

over the group actions, and this sequential method imposes a less heavy memory burden since the operation is applied in series per group element. However, although scaling is quadratic w.r.t. the number of group elements and the domain size for weight-sharing, we mitigate this issue by use of the typically lower-dimensional support of convolutional filters (i.e., and ), rendering our approach practical for a wider range of settings.

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn contrast to many works that impose strict architectural constraints to parameterize neural networks that are exactly equivariant to a known group, this work considers learning approximate equivariances to unknown groups. This is done using soft weight sharing schemes, where doubly stochastic tensors are learned and applied on canonical weights. The approach generalizes GCNNs, and is shown to learn natural symmetries in experiments.\n\nSTRENGTHS:\n1. Nice background section. In my opinion, it is written much better than other related papers in the area.\n2. The method is a natural generalization of group convolutions, which is developed via the neat perspective of group convolutions as applications of transformations of a canonical weight tensor. It is nice that you can choose the number of \"group elements\", so that e.g. non-group-symmetries can be captured.\n\nWEAKNESSES:\n1. Section on \"Regular representaitons allow for element-wise activations\" is a little out of place and imcompletely justified. The claim that non-elementwise activations \"in practice are not as effective as the classic element-wise activations\" is a strong statement that needs more specific justification.\n2. Direct parameterization of the kernels for each group element is expensive. Empirical runtime and memory analysis would be appreciated, to see the effect of this.\n3. Empirical results are rather limited and weak. Few baselines are considered (see below as well), the baselines already perform well on these tasks, and WSCNN does not improve over the baselines in Table 1. Given that the method does not appear to be too scalable (see above), it is unclear where it would be useful in improving performance.\n4. Other baselines (besides standard CNNs and GCNNs) and ablations are missing. For instance, one can imagine removing the sinkhorn projection onto the doubly stochastic matrices. If we initialize the $\\Theta_i^l$ to be permutation matrices (your fixing of one representation to be identity is related to this), then this may also perform similarly.\n\nQUESTIONS:\n1. How do the learned filters of WSCNN look on CIFAR10?\n2. Do learned representations ever look the same across different \"group\" elements?\n\nLIMITATIONS:\nSome discussion in section 6", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nOverview:\nThis paper claims to build upon a long line prior work on symmetry detection and learning equivariant kernels. [18,19,20,22, 25-31. Eespecially reference [30] which is an identical weight and parameter sharing scheme that learns  and discovers equivariances  solely by row-stochastic entries.  This paper's idea is to enforce both row and column stochasticity through  use of the Sinkhorn operator  to achieve equivariance discovery and applicable to  more interesting data domains such as images.\n\nAdvantages:\n* The paper addresses an important problem of  Kernel equivariance via symmetry detection and weight  in CNN . However its marginal progress over prior work. \n\nWeaknesses:\n* Comparisons with prior work are missing.\n* The quantitative advantage of  double stochasticity from row and column has advantages for metric spaces (images),. However this is not explained nor quantified through comparison.\n* Moreover while ground truth tests are done for some toy problems, the quantitative generalizability of the double stochasticity over single stochasticity are never delineated, nor proved.\n\nQuestions:\nThis paper is identical to  this ICML 2024 workshop paper.   https://scholar.google.com/citations?view_op=view_citation&hl=en&user=gfRkDXEAAAAJ&citation_for_view=gfRkDXEAAAAJ:ufrVoPGSRksC ?\nAlso why no comparisons to even reference [30] ?\n\nMissing References:\n* Several references are missing, including the entire suite of probabilistic symmetry detection and equivariant NN is not discussed. See for e.g.   @misc{bloemreddy2020probabilisticsymmetriesinvariantneural,\n      title={Probabilistic symmetries and invariant neural networks}, \n      author={Benjamin Bloem-Reddy and Yee Whye Teh},\n      year={2020},\n      eprint={1901.06082},\n      archivePrefix={arXiv},\n      primaryClass={stat.ML},\n      url={https://arxiv.org/abs/1901.06082}, \n}\n\nSTRENGTHS:\nThis paper enforces both row and column stochasticity through  use of the Sinkhorn operator  to achieve equivariance discovery.  However it's not clear what quantitative benefits this paper's method makes.\n\nWEAKNESSES:\nWeaknesses:\n* Comparisons with prior work are missing.\n* The quantitative advantage of  double stochasticity from row and column has advantages for metric spaces (images),. However this is not explained nor quantified through comparison.\n* Moreover while ground truth tests are done for some toy problems, the quantitative generalizability of the double stochasticity over single stochasticity are never delineated, nor proved.\n\nQuestions:\nThis paper is identical to  this ICML 2024 workshop paper.   https://scholar.google.com/citations?view_op=view_citation&hl=en&user=gfRkDXEAAAAJ&citation_for_view=gfRkDXEAAAAJ:ufrVoPGSRksC ?\nAlso why no comparisons to even reference [30] ?\n\nMissing References:\n* Several references are missing, including the entire suite of probabilistic symmetry detection and equivariant NN is not discussed. See for e.g.   @misc{bloemreddy2020probabilisticsymmetriesinvariantneural,\n      title={Probabilistic symmetries and invariant neural networks}, \n      author={Benjamin Bloem-Reddy and Yee Whye Teh},\n      year={2020},\n      eprint={1901.06082},\n      archivePrefix={arXiv},\n      primaryClass={stat.ML},\n      url={https://arxiv.org/abs/1901.06082}, \n}\n\nQUESTIONS:\nQuestions:\nThis paper is identical to  this ICML 2024 workshop paper.   https://scholar.google.com/citations?view_op=view_citation&hl=en&user=gfRkDXEAAAAJ&citation_for_view=gfRkDXEAAAAJ:ufrVoPGSRksC ? Is this ok ??\n\nAlso why no comparisons to even reference [30] ?\n\nMissing References:\n* Several references are missing, including the entire suite of probabilistic symmetry detection and equivariant NN is not discussed. See for e.g.   @misc{bloemreddy2020probabilisticsymmetriesinvariantneural,\n      title={Probabilistic symmetries and invariant neural networks}, \n      author={Benjamin Bloem-Reddy and Yee Whye Teh},\n      year={2020},\n      eprint={1901.06082},\n      archivePrefix={arXiv},\n      primaryClass={stat.ML},\n      url={https://arxiv.org/abs/1901.06082}, \n}\n\nLIMITATIONS:\nThere are no attempts on analyzing the generalizability of the operator to varying initial and boundary conditions or Reynold numbers.\n\nOther potentially relevant paper to consider:\nhttps://www.sciencedirect.com/science/article/pii/S0021999123001997\nThis paper uses error-correction in neural operators based on the residual.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a symmetry discovery through learning parameter-sharing in weight matrices. The parameterization relies on relaxing underlying permutation matrices by transforming them as doubly stochastic matrices. In combination with additional regularization, the parameterization can be used to successfully discovery symmetries in data, improving generalization performance.\n\nSTRENGTHS:\nThe method appears to be very elegant in offering a natural relaxation of underlying weight-sharing scheme through Sinkhorn operator. The paper is well-written and provides nice illustrations which guide the reader in understanding the proposed method. Then, the paper provides thorough empirical validation demonstrating usefulness of the approach in practice.\n\nWEAKNESSES:\n-Regularization.\nThe paper proposes a novel parameterization that can be used for symmetry discovery. In terms of objective, the work relies on direct regularization to avoid equivariants solutions, similar to some other symmetry discovery works. It has been shown in prior work that this strategy can have issues, since it introduces an additional hyperparameter which may need additional tuning (thereby is less ‘automatic’). In terms of explaining the methodology, the paper would benefit from some discussion on the role of the used regularizer.\n\n-Scalability.\nThe paper seems to scale quadratically in |X|. Is this not an issue? How does this compare against scaling of alternative methods?\n\nQUESTIONS:\n-Regularization\nFor experiments, it would be helpful if it is clear how the strength of this regularizer is chosen. Cross-validation?\n-Analysis of learned symmetries\nIt would be interesting to better understand what weight-sharing is being learned for CIFAR-10, apart from merely measuring improved test accuracy. Is there an analysis on the learned symmetry?\n-Regularization\nEntropy regularization seems to hurt on CIFAR-10, but improve on MNIST experiments. Do authors have an intuition on why this is the case?\n\nLIMITATIONS:\nThe method proposes an elegant and novel symmetry discovery method. I expect the proposed parameterization to be beneficial to the community. The paper could improve a bit on the objective function side of things (seems to rely on directly encouraging symmetry, like some prior work). Apart from the remaining questions, the paper makes for a strong contribution.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a parameterisation that contains the ability to represent weight tying corresponding to arbitrary (?) group equivariances. In practice it can represent interpolations of weight tying, but it is argued that this is a feature (not a bug!) since strict equivariance is often too strong a constraint to place on a model for it to still fit the training data.\n\nStrict equivariance is known to be represented by a permutation matrix (correct me if I'm wrong), which justifies the need to use doubly stochastic matrices to interpolate. This makes the edges of the parametisation strict equivariances. (For equivariances of discretised continuous signals this is not quite true, as the permutation in continuous space would need to be approximated by some interpolation in the discrete space)\n\nThis leads to a new parameterisation of a weight structure in a layer that can be trained in the usual way. The doubly-stochastic matrices can then be investigated to see if equivariance is actually learned.\n\nThe experiments implement the method, and run on benchmark datasets and synthetic datasets, showing good performance, and somewhat interpretable group structure appearing. It is unclear what the actual point of the experiments is, since there are many reasons to use equivariance, but the experiments are not phrased in terms of this (see discussion).\n\nSTRENGTHS:\nThe problem of learning equivariances is very important, as it would remove a significant difficulty in designing networks with the correct inductive biases. The solution is flexible, as any (?) group structure can be represented by the parameterisation. There is also an elegant solution to the problem of needing a large number of parameters, that will work in practice for image data: Assuming translational equivariance, and only parameterising additional equivariances on the filters that are much smaller than the image.\n\nWEAKNESSES:\nOverall, the paper presents a well-reasoned method to an important problem, and I do believe that it meets the standard for publication at NeurIPS.\n\n**Method & Presentation**\nThe method is well-justified. However, a final summary of what a forward pass through a layer looks like was not given, and would be really helpful. In addition, it would be helpful to have a clearer discussion of how many additional parameters are added (beyond lines 220 onwards), with the architectures that are discussed given as an explicit example.\n\nEssentially, one thing which seems to be the case, but is not explicitly acknowledged, is that this method collapses to just a special parameterisation of weight matrices, where the weights have low-rank combined with doubly-stochastic structure. The low-rank-ness is shown in eq 6. While this is a simplistic way of looking at the method, it does give a helpful alternative view. Making this explicit would help the paper.\n\n**What is the claim of the experiments?**\nThe experiments are the main weakness of the paper. Some qualitative results about the structure of the learned weights are given, which are helpful. But it is not clear what the quantitative claim of the experiments is. Equivariance can help in several ways, e.g. better out-of-distribution prediction, better prediction at low data, or smaller/compacter models. So is the claim that the equivariance inductive bias helps, and it can be discovered automatically? But in this case this is not disentangled from the model capacity. Perhaps a normal CNN would perform better if it were just made larger! This is additionally indicated that the baseline 70% accuracy on CIFAR 10 is low compared to what other non-group-equivariant methods can achieve. This unclarity also exists in the synthetic experiments, where the size of the dataset is not discussed. Low-data experiments could help here, since it's easier to make the model large enough that size doesn't help any longer, which isolates inductive bias only.\n\nThe CIFAR experiments show that making the model larger improves performance. How can we be sure that this is really the benefit of learning equivariances, rather than adding more capacity? Another experiment that is necessary here, is a comparison to a weight structure that does not have the doubly-stochastic constraint enforced on it. This would allow the effect of simply adding capacity to be tested.\n\nAlternatively, a low-data experiment would allow the generalisation capabilities of the model to be tested (e.g. [Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximations](fig 3 in https://proceedings.neurips.cc/paper_files/paper/2022/file/50d005f92a6c5c9646db4b761da676ba-Paper-Conference.pdf)). This could be done on MNIST variants, where currently the differences are so small, it is hard to draw conclusions.\n\nThis same issue pops up in the synthetic experiments. What is the dataset size? If the dataset is so large that all transformed signals are in the dataset, even a fully-connected network would learn the correct function. A low-data experiment is needed to really show that truly an equivariance has been learned that can help to _generalise_. Alternatively, you need to argue a benefit on the basis of parameter count.\n\nIn summary: The claims that the experiment section support are not clear. The field of equivariances is mature enough that the potential benefits have been clearly described, and these need to be clearly evaluated in experimental sections.\n\n**Related Work**\nThe idea of relaxing equivariance by placing a distribution over transformations is older than the papers currently cited. E.g.:\n- [Local Group Invariant Representations via Orbit Embeddings](http://proceedings.mlr.press/v54/raj17a/raj17a.pdf)\n\nThe discussion of \"symmetry discovery methods\" is unclear. What data do these methods need, or what kind of training signal do they use? What kind of predictive improvements do they obtain? Is the goal of these papers the same as those in the previous paragraph? Or, is the way that these methods learn group structure different from those in the previous section? If so, how?\n\nMethods that learn a degree of equivariance on a layer-by-layer basis are relatively new, and it would be good to discuss this explicitly. E.g.:\n- [Residual Pathway Priors for Soft Equivariance Constraints](https://proceedings.neurips.cc/paper/2021/file/fc394e9935fbd62c8aedc372464e1965-Paper.pdf) Finzi et al is not mentioned at all, but does allow partial equivariance in a layer-by-layer way.\n- Reference [26] \"Learning layer-wise equivariances automatically using gradients\" is cited, but only in the context of arguing that overly constrained models suffer poor performance, but not in the context that this paper also discusses how to learn the right equivariance to use. This author has more papers on learning invariance/equivariances that may be relevant.\n\nOverall, the literature review misses a lot of relevant work. The suggestions I gave are off the top of my head and I definitely missed some important papers too. However, it is the responsibility of the authors to put the time and effort into going beyond this to give a more thorough overview.\n\n**Minor**\n- _\"Requiring no prior knowledge of the possible symmetries.\"_ (line 52) It is true that earlier methods (with the exception of [31]) could only pick between groups that were completely specified a-priori. While this paper _in principle_ does provide a parameterisation that can _search_ over a much wider space, this space needs to be limited for scalability reasons, and it was not demonstrated that the method would work reliably without this \"prior\" being added!\n- It would be really helpful to have a full discussion of the impact on the number of parameters, for these specific experiments.Needs a discussion of the total number of parameters\n- Is there a typo in params in table 1? Under \"Params\" should \"103 + 265K\" be \"103K + 256K\"?\n\nQUESTIONS:\n- Can this parameterisation represent arbitrary group equivariances? Is the representation of every strict equivariance a permutation matrix? It would be helpful to be explicit about how general this really is.\n- Am I right in understanding that this method ultimately just parameterises a low-rank weight matrix over different feature channels?\n- Can the benefit that equivariance promises in the low-data regime still be provided by this method when the invariance is learned? Since effectively, you're just parameterising weights in a different way (low rank?). In rotationally equivariant settings in low-data, could these weights not just overfit, rather than learning to rotate filters? Would this not lose an important benefit of equivariance?\n- How important is the doubly-stochastic nature of things? Could you just run an experiment without the Sinkhorn component at all?\n- Can you give a very short (ideally 1 sentence, or a 2-3 sentences) summary of the quantitative claims that are made about the method, that are verified in the experiment section?\n\nLIMITATIONS:\nSee above.\n\nOverall, this is a really interesting idea. I do have some concerns about the evaluation. These concerns are large in the scheme of determining how well this method really works relative to clearly formulated claims, but small relative to typical approaches in the ML community.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-21"}