Group Downsampling with Equivariant Anti-aliasing
Authors: Md Ashiqur Rahman
Raymond A. Yeh
Venue: ICLR 2025 Poster (2025)
Keywords: equivariance
downsampling
signal processing
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Downsampling layers are crucial building blocks in CNN architectures, which help to increase the receptive field for learning high-level features and reduce the amount of memory/computation in the model. In this work, we study the generalization of the uniform downsampling layer for group equivariant architectures, e.g., $G$-CNNs. That is, we aim to downsample signals (feature maps) on general finite groups *with* anti-aliasing. This involves the following: **(a)** Given a finite group and a downsampling rate, we present an algorithm to form a suitable choice of subgroup.  **(b)** Given a group and a subgroup, we study the notion of bandlimited-ness and propose how to perform anti-aliasing. Notably, our method generalizes the notion of downsampling based on classical sampling theory. When the signal is on a cyclic group, i.e., periodic, our method recovers the standard downsampling of an ideal low-pass filter followed by a subsampling operation. Finally, we conducted experiments on image classification tasks demonstrating that the proposed downsampling operation improves accuracy, better preserves equivariance, and reduces model size when incorporated into $G$-equivariant networks

Published as a conference paper at ICLR 2025
GROUP DOWNSAMPLING WITH EQUIVARIANT ANTI- ALIASING
Md Ashiqur Rahman
Raymond A. Yeh
Department of Computer Science Purdue University
Department of Computer Science Purdue University
rahman 7 9 purduue.eeem edu
rayyeh@purdue.ed.
ABSTRACT
Downsampling layers are crucial building blocks in CNN architectures, which help to increase the receptive field for learning high-level features and reduce the amount of memory/computtaion in the model. In this work, we study the general- ization of the uniform downsampling layer for group equivariant architectures, e.g., G-CNNs. That is, we aim to downsample signals (feature maps) on general finite groups with anti-aliasing. This involves the following: (a) Given finite group and a downsampling rate, we present an algorithm to form a suitable choice of subgroup. (b) Given group and subgroup, we study the notion of bandlimited-ng propose how to perform anti-al anti-aliasing. Notably, our method generalizes the notion of downsampling based on classical sampling theory. When the signal is cyclic group, i.e., periodic, our method recovers the standard downsampling of an ideal low-pass filter followed by subsampling operation. Finally, we conducted experi- ments on image classification tasks demonstrating that the proposed downsampling operation improves accuracy, better preserves equivariance, and reduces model size when incorporated into G-equivariant networks.
and
INTRODUCTION
Computer vision models, e.g., ConvNets or ViTs, consist of striding and pooling layers used for downsampling feature map (He et al., 2016; Liu et al., 2022; Dosovitskiy et al., 2021; Liu et al., 2021b; Wu et al., 2021; Yang et al., 2024). These subsampling layers play a crucial role learning the spatial hierarchy of features, building in translation invariance, and reduction computation (Zhang et al., 2023). Concepts from signal processing (Vetterli et al., 2014), e.g., bandlimited-neee and anti-aliasing, have also been introduced design better downsampling (anti-aliasing followed by subsampling) operations (Zhang, 2019; Zou et al., 2020; Vasconcelos et al., 2021).
Given additional prior knowledge, group equivariant ConvNets and Transformers have been proposed to incorporate additional structure into the models (Cohen & Welling, 2016; Tai et al., 2019; Romero & Cordonnier, 2021; Rojas-Gomez, al., 2022; 2024; Xu et al., 2023). These models have guarantees that the output is transformed predictably when the input is transformed. A canonical example is shift-cquivariance in image segmentation, where the output mask is shifted accordingly when the input image is shifted.
Interestingly, subsampling layers are not as common in group equivariant architectures. Most models only subsample over the translation group. One limitation is that existing subsampling layers (Cohen Welling, 2016; Xu et al., 2021) over groups require knowing the subgroup downsample to. That is, there is notion of "subsampled by factor of two". From a practitioner's point of view, it is often unclear how to choose such subgroup (see Appendix $A1.4 for an example). Furthermore, these subsampling layers are not designed with proper anti-aliasing, which hurts the equivariance guarantees (Gruver et al., 2023).
In this work, we propose generalization of uniform downsampling of signals (features maps) on general finite groups with anti-aliasing. We present an algorithm to form a suitable choice of subgroup given finite group and integer downsampling factor. Next, define the sampling theorem and bandlimited for subgroup subsampling of signals on groups. To ensure the signal is bandlimited, propose anti-aliasing operation following the introduced bandlimited definition

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while maintaining equivariance. We point out that our proposed algorithm and definitions intuitively generalize the notion of downsampling based on classical sampling theory.
Beyond the theoretical aspects, conduct experiments to test the proposed downsampling operation. First, we numerically validate the proposed claims. Second, we conduct experiments on the MNIST and CIFAR-IO datasets to evaluate the performance of the proposed downsampling layer on image classification tasks over different symmetries. We show that our proposed subsampling layer selects suitable subgroups for the task of image classification, and the proposed anti-aliasing operation further improves the models' performance both in task performance and equivariance. Our contributions:
We generalize the uniform subsampling operation to signals on finite groups, allowing subsam- pling at a desired rate, yielding signals on subgroups.
We introduce the Subgroup Sampling Theorem and the concept of bandlimited-ness for subgroup subsampling. guarantees the perfect reconstruction of the signal on the whole group from the subsampled signal on the subgroup.
We propose equivariant anti-aliaa aati-aliasing operation ensure the signals are bandlimited before subgroup subsampling. Empirically, we demonstrate the efficacy of the downsampling operation.
2 RELATED WORKS
Downsampling layers (subsampling & anti-aliasing). The idea of subsampling has rooted in striding and pooling as early as the seminal works of CNNs (Fukushima, 1980; LeCun et al., 1999). To downsample high-resolution feature map a low-resolution one, e.g., by factor of two, one can simply discard every other element in a feature map. More recently, anti-aliasing has been incorporated into deep nets, inspired by signal processing, where they propose to blur the feature map before subsampling using a low-pass filter (Zhang. 2021; Karras et al., 2021; Rahman & Yeh, 2024). Later, subsampling has also been extended to groups (Cohen & Welling, 2016; Xu et al., 2021). However, the term "every other" is ambiguous here, resulting in a definition that assigns groups to specific subgroups without adequately addressing the subsampling rate. Additionally, the anti-aliasing operation is not tailored for groups. In contrast, this work addresses these limitations by creating a theoretical foundation for subsampling by a specified factor within groups and proposing an effective anti-aliasing method that extends the sampling theorem, which we discuss next.
The sampling theorem is the basis of digital signal processing, which studies how to sample, inter- polate, and manipulate signals sampled at different rates (Vetterli et al., 2014). The sampling theorem guarantees that bandlimited signals can be perfectly reconstructed given a high enough sampling rate, This idea has been extended to graph signals and the field of graph signal processing (Chen et al., 2015b;a). The sampling theorems for the cyclic and abelian groups have also been studied (Dodson, 2007; Faridani, 1994; McEwen al., 2015; Napolitano & Spooner, 2001; Vaidyanathan & Kirac, 1999). These works generalize the discrete Fourier transform to discrete groups but do not consider a generalization for all finite discrete groups. Different from these works, we present generalization of the sampling theorem for any finite groups, propose downsampling layer, and show how the layer can be incorporated into group equivariant deep-nets.
Equivariant deep-nets. Incorporating equivariance into deep-nets has been found to be an effective approach designing deep nets (Cohen & Welling. 2017; Bekkers et al., 2018; Worrall & Welling, 2019; Yeh et al., 2022) across many applications in multiple domains, egg., sets (Ravanbakhsh et al., 2017; Zaheer et al., 2017: Hartford et al., 2018: Yeh al., 2019a; Liu al., 2021a: la; Rahman et al.. 2024), graphs (Maron et al., 2019; Liu et al.. 2020: Ych et al., 2019b; Morris et al., 2022; Liao & Smidt, 2023; Du al., 2023), etc. We foresee that our proposed downsampling layer can be incorporated into this rich literature of group equivariant architectures to build more effective and efficient models.
PRELIMINARIES
We now review the necessary background and definitions. For further details, please refer to $A1.
Downsampling of sequences. Given a subsampling factor and a signal a E IRA, the subsampling operation is defined as
Subn IRA
R LN/RJ
AA Zt, where (ubr((:)[m]] A ae [Rn).
(1)
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When we subsample signal following Eq. (I), it can often result in distorted signal due to aliasing. To avoid this, an anti-aliasing filter is used to remove feghfrequency content, i.e., to obtain bandlimited signal, before subsampling. An ideal anti-aliasing filter, denoted as h, is used to remove all frequency content above the Nyquist frequency (Shannon, 1949). In summary. the ideal downsampling can be expressed as anti-aliasing filter followed by the subsampling operation as:
Dwnn(x) [m] (x h) [Pn), where F(h)[i] if fxyquist else O. Here, :R C denotes the discrete Fourier transform.
Remarks: Subsampling a finite sequence involves retaining the signal at every R-th factor. At glance, is not obvious how generalize this subsampling strategy to finite group G. As G set, there is no notion of "every Rt element. Naively sorting the elements and applying the subsampling for sequences would also not work, e.g., the subsampled set may not be subgroup.
G- G-Equivariance. In deep learning, imposing equivariance on the layers is often desirable. We say a linear map (layer) W E R. xn is equivariant with respect to group G with representation Pu G 1 GU) and Por - GL(U") with U C R" and U C ER' if W (e Po. (g) Vg E G.
(3)
Generating set. A subset S of group G' is said to be the generating set if any element E G can be expressed as a product of the elements of We use the notation = (S) to denote that G is generated by S and assume identity element e E S. The set S is called the minimal generating set when (s}) * Gss E S, i.e., every element of S is necessary to generate the group G. Note, a group can have more than one minimal generating set. We call the ht power of an element E S non-redundant sh cannot be expressed as product of the rest of the generating elements S*{{}
when k *
Cayley graph. To better understand the abstract structure of a group, one approach is to represent as graph, namely, Cayley graph. Given group G and its generating set S, a Cayley graph I(G. S) consists of vertices V and edges F. The vertices correspond to each element E G. and there exists an edge (a, E, if there exists E S such that b = a In the directed Cayley graph, edges are directed from a to b. Any element E G can be represented as a path on the I'(G, S) starting from the identity node e.
Fourier transform for finite groups. The notion of Fourier transform has also been studied on groups (Folland, 2016; Stankovic et al., 2005). For finite group G. let G be the set of complex unitary irreducible representations (complex irreps). We denote the dimensionality of an irrep & E G as d. such that (gg)) E xd, Vg G. The Fourier transform of a square- -itterrintegeable function (G) is
f(P!TT)
1
E Vd..f (g) Vips & and 1 S m, S de.,
G
gEG
where (g) denotes the entry at mh row and n column for matrix (p)(g) Next, f(yp"=)) denotes the Fourier coefficient corresponding irrep component ppt.. Similarly, the inverse Fourier transform on a group can be expressed as
f(g) = Σ Σ
"(ppTr) Vd..v." (g).
(5)
PE magdee
where we denote the set of orthonormal Fourier basis as { d.p lpp Ô and m,n < dp: following the Peter-Weyl theorem (Peter & Weyl, 1927). For real-irreps, the orthonormal basis set is constructed by only taking non non-redundant columns of yps (see Supp C of Cesa et al. (2021)).
4 METHOD
In this work, consider signals X6 A f + Rd) to be an unconstrained real- valued function over a finite group G. For readability, we describe the content with d = 1, which can be easily generalized. A group element g acts on the space X6 via a regular representation Pxo ie.. (Pxo (g)))(u) = -u) Yu E G. Our goal is to design downsampling operator
Dun. X' + Nov, which resamples the signal on a group G to be on a subgroup G C G.
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This involves addressing the following: (a) Given a group G and a subsampling rate R, what is an appropriate subgroup G (b) Given group and a subgroup, what is the notion of bandlimited-ness to guide the design of anti-aliasing? We answer these questions by proposing downsampling op- eration that generalizes the existing notion of sub- sampling ($4. 1) and sampling theorem ($4.2) from sequences finite groups.
Algorithm Uniform group subsampling
1: Input: Group G, Generators S, subsam-
pling rate R, generator &a
2: Output: Suhsampled group G
3: // Get directed Cayley graph
4: V. E + DiCayl((C
5: E' t E.oppy))
6: for each
V do 7: // remove generator 8d
4.1 UNIFORM GROUP SUBSAMPLING
E' F'.remove(UU. sa))
9: // add generator s Rt
A natural generalization from subsampling of se- quences in Eq. (1) to signals on a group is to keep the signal on subgroup G and discard the rest:
10: E" D'.add((u,v x)
11: end for
12: / BFS tzaversal from e
x² (xb)(x) with [g'] xوsا
Vg' E GO,
(6)
13: Q 2
14: G+ +
where the downsampled signal is denoted by x²
15: Q.enqueue(e)
G+ + R. However, it is not obvious how to obtain such G and how to relate it to the rate R.
16: while Q & Ø do
17:
n f m+-QQae -Qd.qquuuee)
18: G G-add((n)
Given a group G and subsampling rate R. we propose the uniform group subsampling, which re- turns subgroup G.. Our subsampling algorithm
19:
for each (n.m) E E' do
20:
if & Q then Qenqqueee(m) end if
intuitively generalizes from the traditional subsam-
21:
pling and is guaranteed to return a subgroup under
22:
mild conditions (details in Clm. 1). Our approach
23: end for
breaks subsampling into two parts: subsampling
24: end while
25: Return G+
on group for a specific generator (Alg. 1), and how to choose the generator.
The key idea behind Alg. is to leverage the structure of the Cayley graph to perform the subsampling. Consider a generating set S = {s1, ... 8n } for a group G = (5). Let each generator s; E S to and such order always exists (Isaacs, 2009). We view the uniform subsampling of G by a factor of R for generator sa is to uniformly discard elements along the path the Cayley graph of G. This can also be viewed as adding the generator A to the generating set S while removing the generator sd In Example 1, we illustrate the proposed Alg. applied to sequence.
have an order o;, i.e., 5.
(e, s2 s2 ................................ $d -d-1))
Example 1. Discrete-time periodic signal of period 4. The domain corresponds to the translation group on a periodic 1D grid of size 4. with the generator 1 representing discrete time translation (At). The group action is addition modulo 4, indicating periodic time shift. Its Cayley graph is shown in Fig. (left). When downsampling by a factor of 2, the generator Ât combines to 2At. Observe that this is equivalent to the subsampling in Eq. (1) by discarding every other element.
RM
25
2A1
2
A
A
a
(b)
(c)
Figure 1: (a) Cayley graph of the group with generator At. (b) Edges corresponding to the generator At 1 are removed (dotted edges) and new edges corresponding to the element 2At are added. (c) The resultant cyclic subgroup of size 2 obtain by the traversing the new graph from node O.
Equipped with the intuition, we now introduce two lemmas before going into the conditions and show why Alg. I returns G+ that is subgroup of G.
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Lemma 1. For the Gt returned by Alg. 1, G. ifand only if can be expressed product of the elements of the set St =SS(+a)) U {
Lemma 2. For the set 5h in Lemma I, each element 8; E E = sT G
Please see $A3.1 and $A3.2 for the complete proof of the lemmas. With some mild assumptions. using the above lemmas, we can show that the set G+ returned by Alg. is subgroup of G (see $A3.3). To guarantee that we are indeed downsampling. additional conditions are required such that G+ is a proper subset of G, i.e., the size of G+ is smaller. Specifically, we need conditions to ensure that the discarded group elements cannot be regenerated from the remaining ones.
Claim 1. If St { k Zt and k. mod & 0} are non-redundant powers of 8ds oa mod r = O, and the elements of St can not be represented as product of the elements of the left cosets of the subgroup Gsub = (S/ (8a)) generated by the set { n E Znj then Alg. returns a proper subgroup G G.
Proof. We show that G+ forms group by verifying closure (using Lem. 1), the existence of inverses (using Lem. 2), and that associativity and identity hold by construction. We then prove that G4 is proper subset of G by showing that, under the assumptions, elements can only be discarded in Alg. 1. The formal proof is provided in $A3.3.
Clm. imposes conditions that restrict the regeneration of discarded elements by Alg. 1, ensuring a proper subgroup. For a better understanding of the implications of this claim, we provide a visual illustration in $A4. With Alg. I, we can subsample a group given specific generator. If there are multiple generators, then different subgroups can be formed. We now discuss how to choose among these subgroups.
Choice of subgroups. The choice of subgroup matters. Choosing generator sa with small order to subsample may lead to the complete exclusion of transformations associated with it; see Example 2.
Example 2. Subsampling of dihedral group Dz. Here, we illustrate the effect of the choice of generators while subsampling the group Ds (s,ris² r e. sr r)s). While subsampling by factor of we can subsample along the generator resulting in a cyclic subgroup of rotation Ca. Or, according to our proposed algorithm, we can subsample along v, resulting in subgroup D4.
Ds
C4
Ds
D4
s.bcamplingty
Figure 2: Subsampling group D& along the generator (on left) and r (on right). The edges
corresponding the subsampling generators dotted in the Cayley graph.
Based on this intuition, propose a heuristic for selecting set of generators D, to subsample G with sampling factors R, along each generator in D,. Given the subsampling rate R, we decompose it into prime factors, i.e., R R1 R2 R3 ... sorted in descending order. For each R, starting from 1, we select the generator with the maximum order satisfying the constraint outlined in Clm. 1. Subsampling by the factor R can be conceptualized
sequential subsampling, each by R, The algorithms for a generalized approach to subsampling and their time complexity analysis are provided in $A5. With subsampling defined, we will next generalize the notion of bandlimited-ness and propose an equivariant anti-aliae unti-aliasing operator to signals on a group.
Remarks:
The proposed algorithm and heuristic offer general framework for uniformly subsampling subgroups from any finite group, extending the concept of sampling rate to groups. The heuristic seeks maximize the number of generators in the subgroup. practice, choosing the subgroup is key hyperparameter influenced by the application and may require domain expertise.
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4.2 THE SUBGROUP SAMPLING THEOREM FOR SIGNALS ON GROUPS
In multi-rate signal processing, the sampling theorem states a sufficient condition (bandlimited-ness) on signal such that perfect reconstruction rate (Vetterli et al., 2014), ie., how to sample and interpolate between finite-dimensional vectors. In this section, we propose sampling theory for signals on finite groups, i.e., condition that allows for perfect reconstruction from subgroups, and an anti-aliasing filter to ensure that the signal satisfies the condition. We now establish a vectorized notation of the signal to aid the discussion.
be achieved given the signal sampled lower
Recall, we are considering a signal z XG - {o G R). where G is finite set with size N, then x can be equivalently expressed by a fintte-dimensional vector x E R. such that x(i] x x[g:). where 9: denotes the 1thh element of the group G in an arbitrary fixed order.
Using this notation, the Fourier transform for a finite group in Eq. (4) can be expressed as a matrix multiplication x F where & C.V denotes the Fourier coefficients. Similarly, the inverse Fourier transform can be expressed as x = F &. Note, JG and Fa are orthonormal bases.
Next, the sampling operation in Eq. (6) and the interpolation operation can be expressed as matrix multiplications:
Sampling: x = Sx,
Sampling followed by Interpolation: x = Ixx ISx,
where S E RR MxN (with M < N) is the sampling matrix and I E RN denotes the interpolation matrix. A perfect reconstruction is achieved when x = x, which is not true in general. Eq. (7) describes the standard setup utilized for deriving the Sampling theory for signals on different domains (Vetterli et al., 2014; Chen et al., 2015a)
We now define the sufficient condition, i.e., "bandlimited-ness" for signals on groups where perfect reconstruction is possible from signals on the corresponding subgroups.
Bandlimited functions for subgroup subsampling. Our main insight is based on the observation that for any bandlimited function x we need to establish map .M E N M from the Fourier coefficients of the subsampled signal x+ & Fouxt to the Fourier coefficients &, which results in the following dependencies between
x = Mi*
x= FM+
xFSFF'
(8)
Combining Eq. (8) and the fact that xt = Fol xt, we establish the following relationship between M, S and the Fourier bases:
= M) SB.
Eq. (9) can be informally viewed as "choosing" set of vectors B & FG M defined on G such that when subsampled to the subgroup G+, they generate the Fourier basis for the subgroup G I. Consecutively, we define the interpolation matrix as I = BFG
We now state our proposed definition of bandlimited signals in the context of subgroup subsampling. G. if the Fourier coefficients & are
Claim 2. Subgroup Sampling Theorem. For any signal in the l-eigenspace of M M( 'Mt then it can be reconstructed perfectly from the subsampled signal xt on Gt. The superscript i denotes the conjugate transpose. Proof. To prove the claim, we show that
x = M)
x B=B
x= PMM
(10)
Here, PM A B(B denotes the projection matrix to the column space of B a F. M. This means that x is in Span(B)) (B), i.e., we can express x Bx, for some set of coefficient vector X.- Perfect reconstruction from the subsampled signal x² is now possible, i.e.,
Ix (BF (SS) =
= FF.o.. = B&. x.
(11)
The complete proof is provided in SA3.4.
To provide some intuition, let's study how this definition applies to Cyclic groups
'The construction of such an M for arbitrary group G and its subgroup G+ is nontrivial, the irreps of the group and the subgroup that constitutes the corresponding Fourier bases often differ in dimensions.

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Example 3. Bandlimited-ness for Cyelic Groups. For real-valued functions over the fi- nite cyclic group CN. the real Fourier bases consist of the constant function VN and (VV cos 2 & + sin 2 N k L Vn [2/NE] where represents the frequency, and n E Z/NZ represents the elements of CN- If N is even, there is an additional basis VN cos 2T 2. Assuming the Fourier coefficients are arranged an ascending frequency and uniform downsampling by factor of (with N mod 2 = 0J, we have
VN
I
M=v3 /2
(12)
ON
where 0g is a zero matrix of size NY x z. as the Fourier bases of Cz are formed by sinusoidal of lower-frequencies. The corresponding M &NNxN is:
{
ifi= j and
Mij
(13)
otherwise
The vector & lies in the Thee--eiiee leigenspeace if = 0. i N, aligning precisely with the conventional
concept of bandlimited-ness.
finite group to be bandlimited with respect to a given M that satisfies Eq. (9). To ensure that the signal is bandlimited before subsampling. the projection matrix M to ensure that the signal satisfies the condition in Clm. 2, i.e., perform an ideal anti-aliasing. However, is easy to observe that the M is not unique. While many M achieve perfect reconstruction, they may be suitable for feature learning. Specifically, the anti-aliasing operation should be equivariant to group actions and preserve some notation of smoothness. We now discuss how to find such an M.
Remarks: We have now defined what it means for signals
Equivariant anti-aliasing operator. We denote the ideal anti-aliasing operator PM in the Fourier space as PMM= & FaP Our goal is to find a M that achieves perfect reconstruction, performs an equivariant anti-aliasing operation, and extracts smooth features. We formulate this goal as an optimization problem:
2 +A1 (Diag(Fo "F"' (14) M = arg min ( Tvec((M)
Equivariance Objective
Smooth Selection Objective
subject to Fol =SFG M (Perfect Reconstruction Constraint).
Here, À >> is hyperparameter balancing equivariance and smoothness, the superscript denotes the elementwise absolute value, Diag returns the diagonal elements as a row vector and the details of T and L are described below.
To be an G-equivariant the anti-aliasing operator, PM needs to satisfy the following equivariant constraint:
PMPro (g) Pxx PM Ag E G
(15)
Here, we describe the equivariance constraint in the Fourier domain where Pxg (g) corresponds to the action of the group G on the Fourier coefficients formed by the direct sum of the corresponding irreps (see 1A1.2 for details).
Next, Mouli & Ribeiro (2021) show that linear operators that are contained within the l-eigenspace of the Reynolds operator T corresponding to the tensor product representation
PxpONa Pxx (g) & Pxx (g 1)T
(16)
satisfy Eq. (15), i.e., are equivariant, where
T a 1 Σ
gEd PNO (g) Pro 1g
(17)
Hence, the equivariance constraint of P/M can be written as vec(P Tvec(P, vM) (see $A1.3). Fi- nally, we relax this equality condition penalty term to form the equivariance objective in Eq. (14).
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Next, the smooth selection objective is designed to prefer smoother basis functions in constructing the bandlimited subspace. To quantify the smoothness signals over groups, we view them as functions over their corresponding Cayley graphs. We adopt the notion of smoothness from graph signal processing, namely, the Laplacian quadratic form (Dong et al., 2016: 6; Shuman et al., 2013) the smoothness measure. The Laplacian quadratie form for function f on G can be defined as f' where L is the Laplacian of the Cayley graph I'(G, S). A smaller value indicates a smoother function. Intuitively, the smooth selection objective can be viewed as penalizing the Fourier bases by their Laplacian quadratic form weighted by their corresponding elements MM
Finally, solve the constrained optimization problem in Eq. (14) via Sequential Least Squares Programming (Kraft, 1988) to obtain M which defines the bandlimited-nee and a corresponding anti-aliasing operator PMM
Anti-aliased G-CNN. In group equivariant CNN (Cohen & Welling, 2016), the input is first trans- formed to functions/features over the desired group. When performing subgroup subsampling, our designed subsampling and anti- anti-aliasing operator are applied to these functions in the group. We discuss this operation detail Appendix $A6.
5
EXPERIMENTS AND EVALUATIONS
Table 1: Empirical Validation of Claim 2. We
report the recon. error with / (and without) the anti-aliasing operation. Anti-aliasing achieves zero recon. error up to numerical precision.
5.1 EMPERICAL VALIDATION FOR CLAIM 2
We validate our theoretical findings in Clm. 2 by nu- merically checking the recovery of bandlimited func- tions after subsampling. We generate random signals x defined on dihedral group D2n = (s, r|s² = r" (sr)² = e) and cyclic rotation group Cn (r|r" e). sampling each value from the standard Gaussian (O,, 1). We consider subgroups G then apply the proposed downsampling technique: project x onto a bandlimited subspace by x PMx (anti-aliasing) and obtain x² restricted to G+ using S (subsampling). Lastly, we interpolate the downsampled signal the original group using x² = Ixt.
Group Subgroup
Sub. R.
Recon, Err.
D14
2
13723-13/3.8
D28
C44
2
6.54c-13/4.0
C7
9.48e-14/5.2
Di0
4.10e-11/3.3
D20
C10
2
3.03e-11/34
D1
2.78e-14/41.
C15
2
5.18c-13/41
C30
C5
9.54c-14/5.9
In Tab. 1, we report reconstruction error, defined the norm difference - 1 bandlimited signal (x and the interpolated signal x²). We observe that the interpolation operator successfully reconstructs the bandlimited signal. further study the proposed anti-aliasing operator, we visualize its response to the unit sample function da(g), where dG [g] = 1 if y and ( otherwise. This response to &g represents the smoothing filter used in anti-aliasing. In Fig. 3, we illustrate such filters. We observe that for the downsampling of the eyclic group (Ci6 to Cs). the filter is reminiscent of the sinc function (Fig. 3), which is used in an ideal low-pass filter for sequences. This further illustrates the relation of our anti-aliasing the classic anti-aliasing sequences explained in Example 3.
between the
Remarks: In practice, ideal anti anti-aliasing operators are often approximated. For instance, the Gaussian blur filter is commonly used to smooth signals, approximating the sinc function, which has better empirical advantages. Building on our theorem, there is potential for developing more efficient. smoothing filter directly in the group ("time") domain.
5.2 IMAGE CLASSIFICATION
We apply the proposed subgroup selection and anti-aliasing operator to equivariant CNN architectures. Note that, to the proposed anti-aliasing filter PM in deep nets, only need perform the optimization in Eq. (14) only once before training a model.
Experiment setup. As in prior works (Cesa al., 2021; Cohen & Welling, 2017), study the effects of subgroup subsampling and anti-aliasing on group equivariant classification models using the rotated MNIST (Deng, 2012) and CIFARIO (Krizhevsky et al., 2009).
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xc
Sr1
20
C16 Css
D16 C4
D16 to Ds
Figure Visualization of the smoothing filter (P (PM())) used in the anti-aliasing operation for subgroup subsampling. The vertical bar corresponds to the value of the filter at each node, with the downward bars indicating negative values.
Table 2: Performance of G-equivariant models on Rotated MNIST and CIFAR-10 at different subsampling rates R and with/without anti-aliasing filter PM- under the continuous rotation and roto-reflection symmetry (2)))) Sub-group subsampling with anti-aliasing improves both equivariance and accuracy.
Param.
Sym. (SO(2))
Sym. (O(2))
R
PM-
x103
AcC,a aug
ACC1as
ACCorbit
Legut
ACC,o aug
AcC1es
ACCorbit
Laqut
323.11
0.9767
0.8234
0.8346
0.058
0.9752
0.8253
0,8496
0.039
194.09
0,9743
0.8007
0.8106
0.056
0.9774
0.6878
0,5660
0.092
MN 1ST
2
194.09
0.9773
0.8301
0.8358
0.049
0.9807
0.6976
0.5749
0.091
151.08
0.9674
0.7762
0.7907
0.057
0.9731
0.8044
0.8316
0.046
3
151.08
0.9731
0.8057
0.8173
0.047
0.9724
0.8251
0.8451
0.037
4 129.57
0.9831
0.6283
0.5052
0.109
0.9810
0.6614
0.4816
0.109
129.57
0.9827
0.6547
0.5219
0.093
0.9806
0.6978
0.5006
0.098
549.33
0.6934
0.4253
0.3708
0.322
0.7251
0.4463
0.3867
0.265
291.29
0.7060
0.4659
0.4096
0.398
0.7448
0.4757
0.3310
0.555
CIFAR 10
2
291.29
0.7088
0.4868
0.4279
0.336
0.7418
0.4720
0.3274
0.460
205.27
0.7006
0.4337
0.3766
0.549
0.7249
0.4210
0.3674
0.478
3 205.27
0.6945
0.4472
0.3876
0.379
0.7117
0.4794
0.4197
0.411
162.26
0.7075
0.4275
0.2866
0.625
0.7590
0.5205
0.2921
0.607
162.26
0.7000
0.4536
0.3091
0.439
0.7525
0.5425
0.3017
0.550
To rigorously examine the impact of subsampling the rotation and roto-reflection symmetry preservation, we remove the digits '9', '2', and from MNIST following Wang et al. (2023). These digits disrupt the symmetry assumption group action. For instance, the digit '6" overlaps with '9' under a 180 rotation. The same is true for '2% '5' and '4'''' under the roto-reflection group.
the dataset that the labels remain unchanged under the
While we consider the symmetry of the image data to be continuous rotation//oto-rrellccioo (20))) note that feasibility which matches our theoretical assumptions. For MNIST and CIFAR-IO, we train on 5k and 60k training images, and test on images on different levels of transformations (see $A7 for details).
the discrete C24 and D24 equivariant CNN for computational
Evaluation metrics. We propose evaluation metrics to measure the equivariance and classification performance. Given deep-net H. we use the features of image æ to measure equivariance error
H( (g)=) Pace (g) H(x)1
CEqui (x)
Σ SEG
(18)
|H(x)
The representations P.. and Po.: correspond to the group action on the input and output space of the deep-net H. In the context of image classification, P.: represents the action of 2D rotation (and flipping), while P.t remains as an identity, i.e., invariance. Specifically, we use the pooled features from the final equivariant convolution layer the invariant features (Weiler et al., 2018).
For the classification performance, we consider three accuracy metrics for evaluating the model performance under different degrees of equivariance:

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Table 3: Impact of subgroup selection subgroup sampling on a 3-layer equivariant CNN. "g" indicates selection based our method. Our algorithm improves performance for various sampling rates.
Group Sub. Subgroups
ACCao ang ACC
ACCorbis
Dus C12 Co
0.9703
0.6215
0.6128
D24
1,2,2
D24 Dr2 Ds*
0.9726
0.6539 0.5489
D21 Cs Ca
0.9766
0.5244
0,4596
D24
1,4,
D24 Do Do*
0.9767
0.6272 0.4860
D2s C1a Cia
0.9742
0,5852
0.5191
D2s 1,
D28 D14 D14"
0.9786
0.7085 0.5792
1. ACCao augi The accuracy of the model on the original (un-augmented) dataset.
2. AcC1oc: The accuracy of the model on the "locally augmented" dataset.
3. ACCorbit: The accuracy of the model on the full (SO(2)/0(2)) orbit of the dataset.
The orbit is constructed by taking all 10 rotations, and for local augmentations, we report random rotations within +00 of the test set.
Results. In Tab. 2, we report the results for MNIST and CIFAR-10 datasets under SO(2) and ((2) symmetry. We report the average over 3 runs. For all models, the standard deviations are: Accao aug < 0.001, Accorbit and Acc1oe < 0.01, and Lequi < 0.004.
Overall, we observe that subgroup subsampling significantly reduces the parameter count of the equivariant models. However, increasing the sampling rate (e.g., R 4) disrupts the strict equivari- ance constraint, leading to a higher equivalence error (Lequi). This manifests as a decrease both AcCorbit and ACC1ec. while increasing the accuracy on the original test set, ACC,o aug. Next, incor- porating our unti-lliasing operation mitigates the invariance error and achieves higher AcCorbit and Acc1o.. Notably, lower sampling combined with appropriate anti-aliasing significantly reduces parameter usage while maintaining comparable or even surpassing the accuracy and equivariance achieved with the full equivariant models.
We provide additional results of our model on STL-10 (Coates et al., 2011) dataset, where we also observe similar performance gain (see Appendix $A2.2).
Ablations. In Tab. 3, we provide the ablation of the proposed sub-group selection heuristic. For 3 layered equivariant CNN and different sampling rates different layers of the models, we report the accuracy metrics for different choices of subgroups. We observe that for different symmetry groups different sampling rates, our proposed subgroup selection improves the performance in most cases,
Furthermore, in 1, we demonstrate index selection by Xu et al. (2021) can be used with our technique. The results show further performance improvements and confirm that our method can easily be incorporated with existing techniques.
Limitations. As this a theoretical paper, we fully acknowledge that our experiments are limited to small-scale datasets and models. These experiments are meant to study and demonstrate the potential of the proposed framework. Our proposed downsampling layer currently operates on finite groups rather than continuous ones. The time complexity of the subgroup selection algorithm scales quadratically, in the worst case, with the number of edges, [E], in the Cayley graph (see $A5).
CONCLUSION
We propose uniform subgroup downsampling for signals on finite groups with an equivariant anti- aliasing operation. We generalize the uniform subsampling operation to groups and propose subgroup selection method based on maximizing the number of generators. We then extend the sampling theorem to subgroup subsampling, generalizing the notion of bandlimited-ness and anti- aliasing to groups. We apply these theories to equivariant CNN and empirically show that models with subgroup subsampling can achieve comparable even better performance compared to full equivariant models. In summary, we believe our developed theory would serve as the foundation for future research in equivariant deep nets and signal processing on groups. We particularly excited about how to find an "optimal" subgroup for a given task and how to design more effective anti-alian anti-aliasing for signals on groups that would build top of our framework.
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ACKNOWLEDGMENT
The authors would like to thank Renan A. Rojas-Gomez for providing feedback on the draft version of this work, which helped to improve clarity.
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APPENDIX
The appendix is organized as follows:
In $A1, we present review on group theory.
In $22, we present the results for incorporating the equivariant index selection operation with our proposed downsampling technique.
In $A3, we provide the complete proofs of the Lemmas and Claims in the main paper.
In A4, we provide the illustration of the implications of Claim 1.
In $A5, we provide further generalization of the approach and how to check whether given group satisfies our theoretical assumptions.
In $A7, we document additional implementation details. Code is also provided in the supplemental materials.
A1
GROUP THEORY PRELIMINARIES
A1.1 GROUP
Group a set equipped with an operation (-) that maintains the following properties:
Closure: Va, E GE
Associativity: Va, b, G,a- (,a-(b-c) (b-c) = (a-b))- b).c
Existence of Identity: 3e G Va E G, a
Existence of Inverse: Va E G. a:a-a-lze.
H is subgroup of Gf is known as the order of the group. And the groups of finite order are called finite groups. For any subgroup S G, the left coset generated an element right coset is denoted as Hg (hg: h E H).
G and H satisfies all the group properties. The cardinality of the set G
denoted as gH {gh H} and
Discrete Rotation Group (C,,). The discrete rotation group Cn = (r y e) eyclic group representing rotations by integer multiples of 360 degrees.
Dihedral Group. Dihedral group Dzn = rs,r = r" = (sr)² = e) is the group of symmetries of a regular n-sided polygon, with rotations by integer multiple of 360 and horizontal reflection.
General Linear Group. The general linear group GL, IF) is the group of all invertible matrices with entries from the field F. GL(V) denotes general linear group on the vector space V.
Minimal Generation Set. In general group G can have multiple minimal generating sets. For example, for any cyclic group CN generated by the minimal generating set S = {r), S' = {r-} is also minimal generating when and N are relatively prime.
A1.2 GROUP REPRESENTATION
Linear Group Representation. A linear group representation of a group G on a vector space U is a homomorphism from G to the general linear group GL(U). This can be written
p: GL(U),
(A19)
where for each g E G.
p(g) an invertible linear transformation on U.
The map p must satisfy the following properties:
p(gh) = (()) for all h E G.
p(e) = Ir. where I denotes identity transformation on U.
p(g) is an invertible linear transformation.
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The dimensionality of representation p is equal to the dimensionality of U and written dp-
The trivial representation of group G on a vector space is a representation p such that p(g) = ly for all g G. In other words, every group element acts as the identity transformation the vector space U.
The regular representation of finite group G on the vector space FIG] with basis indexed by elements of G act on it by permuting these basis elements according to the group operation.
Equivalent Representation Two representations p1 and p2 are equivalent iff pr TTpTT T-2T-1 for some change of basis T E GL(U).
Direct Sum of Representations Let P1 : G GL(U)) and P2 G - GL(U2) be two representa- tions of group G on vector spaces U1 and U2 over the field F. The direct sum of and p2 is representation P1 @ p2 : + GL(U, @ U2) defined by:
(pl e (2) (1) p2(g) for all g E G,
where Ui @ U2 is the direct sum of U1 and U2. and the action on U1 @ U2 is given by:
le @ p2)(g)(u,, U2) =
for all #n U1 and U2 U2.
Irreducible Representation. A representation - GL(U) of a group G on vector space U over field is called irreducible if the only G-invariant subspaces of are the trivial subspace {0} and U.
The set irreducible representations (irreps) of G is denoted as G. All the irreps abelian groups are I dimensional. The irreps of dihedral groups are and 2 dimensional. And the irreps of symmetric group S4 are 1,2.3 dimensional.
Orthogonality Relation and Fourier Transform. Let y be an irreducible unitary representation of G of degree dg. Then the de functions (v /ddpspmmn 1 S m, S dg) form an orthonormal set.
Let G be finite group. Let & = (Lpis...,999 Then the functions
complete set of irreducible representations of G.
{ {vda.ep l E G, mm1mmmn s dp:
form an orthonormal set in the space of complex-vale vaaludd functions over group L(G) de +...+ dZ p, = |G]. In fact, this set orthonormal bases defined the Fourier basis for functions on the group G. The Fourier transform of a square-integrable function f E (.²(G) is
flyp! = G
Σ SEG
flp)f( (g) Viz E G and S m. S dp.
(A20)
where (g) denotes the entry at thth row and th column for matrix (g(()) Next, ((p de- notes the Fourier coefficient corresponding to irrep component pp. Similarly, the inverse Fourier transform on group can be expressed as
f(0) Σ Σ f( ( dep.ppin (g),
(A21)
PIEO mnstl
And if for any g E G the action on f E (C( is defined as [g fl(u) = f(g ((g--u) then the action can be represented in Fourier space as
g
= -
(A22)
where, Pi, f((P)) g f(P)) Cd xdv.
In the real case, irreps can have redundant columns. To eliminate redundancy, an endomorphism basis Co, is constructed to span the non-redundant columns of irrep th The full irrep can be recovered by multiplying these columns with elements C The reverse of this process can also be constructed, giving us the non-redundant columns (see Cesa al. (2021) for details).
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A1.3 INVARIANT AND EQUIVARIANT MAPS
A linear map W E R" xm is equivariant with respect to the group action Pu GL(U) and Per 9 + -GL(U") with U S _ and U' - 1R if
(P Pur (g) Vg NgEGEEE
(A23)
This imposes the following restrictions on the linear map
Per (g) X Pc. ( vec(W) (ecc)))
vec(W) Vg E G
(A24)
where vec denotes vectorization operation converting matrix to vector. The condition in Eq. (A24)
denotes that vec(W) should invariant to action of tensor product representation Pur@, = Pur (g) &
Po TT on the left. For a finite group G, the Reynolds operator (Mouli & Ribeiro, 2021; Mumford
al., 1994) is defined as
T
Σ plg) SEG
(A25)
G
which is G-invariant linear map with respect to the representation p: GL(X)) on vector space X. And to satisfy the condition in Eq. (A24), vec(W) must belong to the 1-eigenspace of Reynolds operator with respect to the tensor product representation Prr acting the vector space U' & U (Mouli & Ribeiro, 2021).
A1.4 ILLUSTRATION CHALLENGES IN SUBGROUP SUBSAMPLING
To illustrate the challenges in subgroup subsampling, we present an example by subsampling the group
Da (e,r, ,,, sr, sr²),
(A26)
which is relatively small group size 6 by factor of 2, i.e., we aim discard every other element.
As can be seen, discarding every other element of the set will generate the subset H = {e, r², sr). We can see that H is not subgroup as the inverse of p2 is missing in the subset; thus, violets the property of a group.
Additionally, the above example, we first assumed an ordering of elements of Dg. However, we can also choose another ordering of the elements
D: fr.e,r² 8, ST²),
(A27)
resulting in a different subset after subsampling. In this example, even with a small group, the set can be arranged in 720 in different ways, creating ambiguity in the process. So, the traditional subsampling operation of naively discarding elements does not apply groups.
A2
ADDITIONAL EXPERIMENTS
A2.1
EQUIVARIANT SUBSAMPLING
The work Xu et al. (2021) proposes to select indexes in a consistent manner that respects a specialized equivariance (see Xu et al. (2021) lemma 2.1) that can work between groups and subgroups, which is equivalent to Chaman & Dokmanic (2021) in traditional subsampling. The work assumes that the subgroup is already provided and does not perform any anti-aliasing. However, the equivariant index selection scheme can be incorporated with proposed subgroup selection and anti-aliasing operator. In Tab. A1, we provide the results on rotated-MNIS, -NIIST (SO(2) symmetry), where we incorporate equivariant index selection with proposed subgroup selection and anti-aliasing operator. We observe that the proposed anti-ul aliasingn operator consistently reduces the equivariance error and improves accuracy. It also demonstrates the wide applicability of our proposed method.
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Table AI: Performance of G-equivariant models Rotated MNIST at different subsampling rates and with/without anti-aliasing filter PM with the equivariant index selection. We can observe that our proposed technique improves performance and equivariance error, showing wide adaptability.
Sub. R.
# Param. 10a
PM
ACCso aug
ACC1ec
ACCorbit
Cequi
2
194.09
0.9733
0.8147
0.8225
0.0545
2
194.09
V
0,9782
0.8288
0.8297
0.0473
151.08
0.9692
0.7717
0.7865
0.1061
3 151.08
V
0,9650
0.7737
0.7833
0.0594
129.57
0.9656
0.6606
0,5602
0.0881
4
129.57
V
0.9703
0.6928
0.5759
0.0761
A2.2 RESUTL ON STL-10
We conduct experiment STL-10 (Coates al., 2011) dataset. This dataset comprises images with a resolution of 96 x 96 from 10 different object classes. We train on 5000 training images with any data augmentation. In Tab. A2 and Tab. A3, we present the result on rotation and roto-reflection symmetry groups, respectively, at different sampling rates. We observe that our proposed method achieves higher accuracy and lower equivariance error compared to naive subsampling. We also observe that, since the images in STL-10 have much higher resolution compared to MNIST and CIFAR-10 and contain more high-frequency details, our anti-aliasing operator provides higher performance improvement.
Table A2: Performnace of G-equivarinat models on STL.-10 dataset different sampling rate R and with/without anti-aliasing filter PM under the rotation (SO(2)) symmetry. Sub- subsampling with anti aati-aliasing improves both equivariance and accuracy.
Initial Group
Sub. R.
PM-
#params
ACCns ang
ACC1oc
ACCorbit
Legus
C24
1.3M
0.54
0.34
0.30
0.16
C24
2
962K
0.60
0.42
0.37
0.16
C24
2
x
962K
0.60
0.40
0.35
0.17
C24
3
831K
0.62
0.42
0.37
0.16
Cz4
3 X
831K
0.60
0.38
0.34
0.18
Table A3: Performance of G-equivariant models on STL-10 dataset at different sampling rates R and with/without anti-aliasing filter PM under the roto-reflection (O(2)) symmetry. Sub-group subsampling with anti-aliasing improves both equivariance and accuracy.
Initial Group
Sub. R.
PMr
#params
ACC aug
ACC1oc
ACCorbit
Lequi
D21
1.3M
0.57
0.37
0.32
0.12
D24
2
V
962K
0.64
0.40
0.27
0.19
D2a
2
X
962K
0.61
0.40
0.26
0.20
D2A
3
V
831K
0.64
0.44
0.39
0.17
D24
3
X
831K
0.60
0.33
0.33
0.17
A2.3 EQUIVARIANCE ERROR PROPAGATION
To further study the effect of the anti-aliasing operator, visualize the propagation of the equiv- ariance error through the model on the STL-10 dataset. The latent functions G-CNN are defined on x G. where G is the rotation dihedral group. To visualize the equivariance error, we pool the latent function over the group G following the method in Cohen & Welling (2016), ensuring equivariance with respect the group action on the input Py. For an input image layer and equivariant G-CNN g, we defined the equivarinace error at each spatial location (i,j) as
E Beg [i [i (Pool,C*(I)bb.) (pg- Pool,9* (
(A28)
||ooll g+ (1)
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where (I) denotes the output of A hidden layer given the input and G. In other words, the equivariant error Eeg j. j] denotes the equivariance error at pixel (i,j). As can be observed in Fig. Al- A8, approach with anti-aliss anti-aliasing shows lower equivariance enror throughout the features at all three layers. Recall, that our experiment uses an architecture consisting of downsampling operation between each of the layers. We also observe that the equivariance error indeed propagates and worsens deeper into the models. Finally, we note that perfect equivariance, i.c., zero error, is not achieved due the boundary pixels going out of scope in the rotated images.
Equivariance Error Layer 2
Layer
Layer 3
Input Image
Without Anit-aliasing
Rotated Image 0 15
With Anit-aliasing (Ours)
Figure A1: Visualization the equivariance error at each layer.
Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 30
With Anit-aliasing (Ours)
Figure A2: Visualization of the equivariance each layer.
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Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 = 15
With Anit-aliasing (Ours)
Figure A3: Visualization the equivariance error each layer.
Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 30
With Anit-aliasing (Ours)
Figure A4: Visualization the equivariance error each layer.
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Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 = 15
With Anit-aliasing (Ours)
Figure AS: Visualization the equivariance error each layer.
Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 30
With Anit-aliasing (Ours)
Figure A6: Visualization the equivariance error each layer.
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Equivariance Error Layer 2
Layer
Layer
Input Image
Without Anit-aliasing
Rotated Image 0 = 15
With Anit-aliasing (Ours)
Figure A7: Visualization the equivariance error each layer.
Equivariance Error Layer 2
Layer
Layer
Rotated Image 0 30
Without Anit-aliasing
Input Image
With Anit-aliasing
Figure A8: Visualization of the equivariance error each layer.
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A3 COMPLETE PROOFS OF LEMMAS AND CLAIMS
A3.1 PROOF OF LEMMA
First, we provide the proof of the lemmas.
Lemma 1. For the set G+ returned by Alg. I, v E G+ ifand only if can be expressed as product of the elements of the set St (S/(sa)) (s
Proof. In the graph (V, E"), there exists path between and some node V iff v G (guaranteed by BFS traversal algorithm). So will be sufficient to prove that in graph (V, E"), there exists path between node and some node if and only if can be expressed the product of elements of S
By construction, each node in directed Cayley graph EV, DiCay(G,,) has out-degree of [S], with each outgoing edge corresponding to an element of the set S. Removing all outgoing edges corresponding to the element sa and adding new element s i.e., E' = E {(a, a 8a) a E Vj {(a, a <8) : E V) maintains the property with respect to S That means each node in graph (V, E') has an outgoing edge corresponding to each element of the set 84,
outgoing edge to each node corresponding the
Let's assume there exists a path from e to node a E V. We denote the path as a list of vertices by {e, (e sa,).-.. (e Sa1 .... Sam-1 Sa.,,) which constructed by picking hops from along the edges corresponding the elements {Sai:.... .Sam-1 Sam. } in order where vj Saz E S4, This implies I1 Saj. i.c., a is generated by products of the elements of set 5
Conversely, let I 8h, such that Vi 8b, E S. Existence of a path from to b demands the exis- tence series hops from along the edges corresponding the elements 8b, 88,, $b... Such series of hops always exists in graph (V, E*) as every node has |S"| out-going edges corre- sponding to each element in S
A3.22 PROOF OF LEMMA 2
Lemma 2. For the set St in Lemma 1, each element 8; E St = 8, E G².
Proof. Let 88 St \ { then sf = ou-2 (as on is order of 84), i.e, oy can expressed as product the elements of 5 by Lemma 1 E Gt.
Now (s = $8 -R Let, = (os 1) and (Rw mod ou) = (Roa R mod ou) = (- R R wwf And, following Lemma 1, (x/f)= E G4.
mod oa). SO, $
A3.3 PROOF OF CLAIM
Claim 1. If S = {sh k E Z and k mod R & 0} are non-redundant powers of Sa- Oa mod r == and the elements of 5h can not be represented as product of the elements of the left cosets of the subgroup Grub = (S/ generated by the set {sa R. n. E Z then Alg. returns a proper subgroup G+ C
Proof. We first prove that G is a group.
Existence of Identity By construction, always member of set G+ start the traversing the graph from node e.
Closure Let a, E G Therefore, by Lemma = 11m So, b II sb, with Vsa, Sb, S4 Now a b= ( sn.) ([I 8b,). i.e, a can also be expressed as product of elements St. So, by Lemma
G
Associativity As G C G. and element of G follows the multiplication table of group G.So, the associativity of (.) operation will hold trivially for elements of G+.
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Existence Inverse element Let, G+ and I group element as S5. 8.n-1 teg And, we can see that u. = ut e. So, u = e By
Se; Sen Su2 ... Sv,, Now construct
Lemma 2, Vi Sag E G and following the Closure property G., Le., G is a group.
Now, we prove that G C G by contradiction. We assume that 3s E 5y such that 5 E
As the elements of S are non-redundant, F d. S+ { ) Additionally, ki mod R & 0 and og wft for some E 2 [ divides oa), BM E 2 IR mod од kg So, AI Z such that (sf)²= st Therefore, the generators for the element s must include o and elements from the 5"4,
not be generated only by the generator 8
Without any loss of the generality, assume that the path from e to s is the shortest among the elements of St n G+. Lets od not be 2 As
=811
Sk2 8kn Sk, such that Vi 8k, E St, Now, 8ky can 8k2
by 8kg 810
d
8k2 8kg -k
Skm
Sk..
where mod + 0 as ky mod O. But sd R E Sh requires one less generator, thus
contradicting our assumption that the path from to p is the shortest among the elements of S nG4,
ks must start and end with A similar restriction is also applicable for 8k., So, the path from to s generators from set S74 Therefore,
h express f
k. d = 91 (s Rnz 92) (s
Rrs (3)....
Rra
q:)
(A29)
where, Vj Qj Gsub with Goub is subgroup generated by Stt, and Vi nę Z.
Next, $d Anm Im for S m S l is an element of the left coset of the subgroup Goub generated by element s/ Rum i.c., Sd Rn.. 4m E {s Ru... .g: E Gsub) and qi is an element of a trivial left coset of
Goub generated by e. Therefore, sd k; is expressed as the product of the elements of the left cosets of Gsub generated by the set {syR n E ZT); which contradicts our assumption.
This means that ds G+ Vs E St and implies that G+ C G.
A3.4 PROOF CLAIM
Claim Subgroup Sampling Theorem. For any signal x on G, if Fourier coefficients & are in the I-eigenspace of M & M(M then it can be reconstructed perfectly from the subsampled signal xt on Gt. The superscript i denotes the conjugate transpose.
Proof. First, show that if
the 1-Eigenspace of MMMMM)-MMM then Span(B) with B A FG M. If x is in the 1 l-eigenspace, then
* = M
(A30)
3F6x FoFe MMM M(M MM- (as FoFo =I)
(A31)
3x Fe.M(M'M)-M''
(A32)
* FF. "c'M (as Fas FG)
*x =Fa' F.'M(''E. "F.M
(A33)
(A34)
* B)BBB) Btx
(A35)
*X PMX
(A36)
Here, PM A B)B BB denotes the projection matrix to the column space of B. Note that the columns of B are linearly independent. As F SB and rank(F=l) M. The rank(B) is at least M. And as B has M columns, they are independent, and rank(B) = M, PM is a valid projection matrix.
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This means that is in Span(B). i.e., express B& for some set of coefficient vector x Perfect reconstruction from the subsampled signal x is now possible, i.e.,
Ix (BFx))(Sx) - (BF = FF = B&c- = X.
(A37)
In conclusion, perfect reconstruction of x is possible from x when x is in the I-Eigenspace of MM
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A4 ILLUSTRATION OF CLAIM
Here is an illustration Claim 1. The claim states that if = { E and mod r # 0} are non-redundant powers of 8d, 0g mod = 0, and elements S can not represented as a product of elements of left cosets of the subgroup Geub (5/(aa) generated by set {sa n E % ] then Alg. returns proper subgroup G+ G. In Fig. A9, we illustrate the claim with group Ds.
Goub (5/(sa))
e
S
r
S
r
T
T
SI
S
D8
ST
Sp.3
T
T
r
ST
S
2
Figure A9: Illustraion of Claim for subsampling Ds by a factor R = along the generator 8d = r. The red-colored nodes denote the set S (r, r3). The green highlighted nodes (e,s) is the Gous (S/r). We can see 5t is nonredundant, and the order of sa is divided by 2. The last part of the claim implies that the colored node must not be reachable from nodes Goub with a hop of 2 denoted in dotted blue line. Which is indeed satisfied with the example shown.
A5 GENERALIZATION SAMPLING ALGORITHM
In this section, we provide an algorithm (see Alg. 2) to check for compliance of generator with the condition in Claim We also provide general sampling algorithm (Alg. 3) that maximizes the number of generators in the subgroup following the heuristics from $4.1.
The Alg. 2 takes O(VV+ [E)) time where the set of nodes and E is the set of edges in the Cayley graph. To choose the generator with the highest order, we need to check for compliance for each of the generators, making the time complexity to downsample by a chosen generator O([S).((V]] (())))))))) The computational complexity can be high for complex groups depending on the choice of the generating set S. Since the sampling algorithm runs only once before training to generate the sampling matrix, efficiency is maintained. Furthermore, for large complex groups, such symmetry groups Sn, the subgroups can be selected based on prior domain knowledge followed by our proposed anti-aliasing operation.
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Algorithm 2 Check-Compliance
1: Input: Group G, Generators S, Generator s, Order of the generator subsampling rate r
2: Output: True, False
3: if mod r + 0 then
4: Return False
5: end if
6: V, + *-DiCay(G,
7: for each v V do
8: E.remove(w, v (())
9: E.add((v, (.00) 82)
10: end for
// graph traversal from e 12: Q+
Geosets + 14: Q.enqueue(e)
15: while Q + Ø do n Q.dequeue()
16:
17:
Geosets-add(n for each (n, m) E' do
18:
m & Q then Q.enqueue(m) 21: end if
19:
20:
end for 23: end while
if = Geosets such that & mod r + 0 then
25: Return False
26: end if
27: Return True
Algorithm 3 General-Subsample
1: Input: Group G. Generators S, Order of the generators O, subsampling rate r,
2: Subsampled Group: Gt
3: E.E + DiCay((,S)
4: G +- G.copy()
5: R + faclorize(r)
6: for i = to R.enggh() do
7: inde: +- NULL
8: for j = 1==1 to S.length() do
9:
// check the compliance of Sli using Alg. (aacck--complaccce[[__S]_/. if (inder NULL OR Oijl < O(indez)) then inder +
10:
R(j) then
11:
12:
end if end if
14:
end for
16:
if indez NULL then
Return NULL end if
18:
19:
// Downsampling using Alg.
20:
G+ + DownsampleeG, S, R[i), Sindde)))
21:
// updating denerating set and order
22:
Slindex) S(inder] R[:]
23:
Olinder] <- O (inder)) [[
24:
end for
25: Return G
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A6 FUNCTION ON GROUPS IN EQUIVARIANT CNN FOR IMAGES
Group Convolution: In the group equivariant convolution neural network the input image f IR (k=1 depending on whether the image is grayscale or colored) is first lifted to the space of roto-translation or dihedral-translatin group (2² x CN or 2² x DN) by the lifting operation (Cohen & Welling, 2016)
* (+)) = ΣE AL V. G, =EZ K
(A38)
where & is the channel index, i.e., fu represents channel of the image, th 22 is 2D kernel, and G' is either cyclic (rotation) group or dihedral group (CN or Dx) for most computer vision tasks. This transformation lifts the image to the desired group by repeatedly applying the transformed (by the action of the group) filter on the image f.
The filter the is a regular convolution filter, i.e., it is real-valued function defined on 2D grid Z². The action of G on the filter de is defined as
[2)))(2) = -h0 Vz EZ.
(A39)
In other words, the transformed filter [0 is defined through the action of the group element g on the = Z², which we directly use in Eq. (A38). In the case of the rotation group, the group element corresponds to angle E [0, 2nr), and the action of the group element on 2 [u, v] E z is defined
cos sin 0
sin 0 cos 0
02 a
(A40)
This is the underlying mechanism of rotating real values function 2D grid (for details, please (Cohen & Welling, 2016; 2017)).
Next, in the group convolution network, the function [f+ 2² G + R is function over 2² x G and is passed on to the following group convolution layers. For the ease of notation, let denote the real-valued function on group Z² G as E, i.e., E G + R. For any E, the group convolution (Cohen & Welling, 2016) is defined as
E <)(
Σ hER
8(h)a(g
Vg 2Z G,
(A41)
where k: x G -> IR is the group convolution kernel. The output is then passed through point-wise non-linearity and followed by more group convolution layers.
We can see that group convolution defined by the action of 2 x G on the function n. This is analogous to regular convolution, where the function k is also "shifted" (transformed) by the action of group elements. For example, the specific case roto-translation group 72 x CN. group convolution is analogous to 3D convolution where the action of the roto-translation group guides "shift" on the filter. Please see Sec. 7 of Cohen & Welling (2016) and Bekkers et al. (2018) for details.
When performing group convolution, we follow the techniques introduced by Cohen & Welling (2016; 2017) and do not propose any modification of the group convolution operations (Eq. (A38) and Eq. (A41)) introduced in the earlier works. For detailed explanation and construction of the group equivariant architecture, we refer the readers to Cohen et al. (2019); Weiler & Cesa (2019); Kondor & Trivedi (2018); Cohen & Welling (2016); Bekkers et al. (2018).
The function E is represented as a tensor of size H x W x [G', where H x W is the resolution of the input image, which corresponds the size of the translation group and |G| the number of elements in the group G.
Anti-aliasing:
Now, fixed translation group element (spacial location) (i,5) E (O,.., .., H} x (0, W} of the tensor, we have function over G. i.e., Vi,,j) E (0, .., H} x (0, W). we have
8(h = i, w = j.d) (d) E R vd e G,
(A42)
with Es,y G R. This function Es,; is transformed according to the regular representation of G. We represent Fa.y vector of size [G], i.e., Fa. E G
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f:
E [f w/ 22 x C4 +
Figure A10: Visualization of function on the group in C4 = {e, (e, r², r3 equivariant CNN. The input image f is transformed into a function over group following Eq. (A38) with some learnable filter v) The resultant function [[ * 1b] a function over 2² C4. Now every fixed spatial location (i,j) E Z. x Zy. we have functions over C4. Elements of one of such functions are marked with dotted circle with corresponding elements of C4
In our work, we perform subsampling and anti- aliasing on the functions Es,, v(. E .., H) x {0, ..., Wj. The anti-aliasing operator PM. can also be represented as matrix of size G |G], ie., PM. E R|G]x|G| and P P9M=(9.9's E R Vg,g' E G. Specifically, the anti-aliasing operation on the function & is defined as
E' = 1;=i,0 i, i j,dd -- = i, w = (m,(dd V(i.s)
(A43)
IEG
with E'(h = i, w- j. (d).
Finally, Eq. (A43) can implemented as matrix-vector multiplication as
& Eij.
(A44)
In other words, the anti aiiiiiliiiinn is matrix multiplication along the group dimension of the tensor representation of E
A7 ADDITIONAL IMPLEMENTATION DETAILS
is essential note that our experiment setup different from that of Weiler & Cesa (2019) and designed carefully to highlight the robustness of the group equivariant model in limited data setting. The evaluation metrics are designed to explicitly measure the consistency of the models under all group actions. Unlike Weiler & Cesa (2019), we do not train the randomly rotated datasets, thereby revealing the actual equivariance property of the model by the architecture design. Also, our designed accuracy metrics, ACCorbit. ACC1oc: and ACCnnaug provide the performance of the model at different granularity under group actions, which can not be obtained by testing the model on randomly rotated tested (Weiler & Cesa, 2019).
For MNIST, we train on 5, 000 training images without any data augmentation and test on 10, 000 images different levels of transformations, For CIFAR-10, we train on 60K images without any data augmentation and evaluate on 10K images. All models consist of 3 group equivariant convolution layers (Cesa et al., 2021; Cohen & Welling, 2016) followed by linear layer mapping to the final logits. The filter size at each layer is 5. When subgroup subsampling is performed, the convolution layer following the subsampling layer is equivariant only to the subgroup. The output of
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the final convolution layer undergoes global-pooling operation (Weiler et al., 2018) to obtain invariant features. For subsampling, roto(dihhedrll)-trann translatonn group, we subsample rotation (dihedral) group and translation group independently. Subsampling along the translation group is equivalent to spatial subsampling and is performed using BlurPool (Zhang, 2019). We set À = in Eq. (14) for obtaining M
Models are optimized using the Adam optimizer and trained using 15 and 50 epochs with batch sizes of 128 and 256 for MNIST and CIFAR-IO datasets, respectively. All the expenses are run on single NVIDIA RTX 6000 GPU.
A8 LATENT FEATURE RECONSTRUCTION
To further investigate the effect of our anti-aliasing operator, we reconstruct the feature on the whole group from the downsampled features the subgroup. is crucial to note that the naive subsampling operation in previous work (Xu et al., 2021) lacks a suitable interpolation operation. Hence, we used our proposed interpolation operator to reconstruct the feature, from the first group convolution layer, for both with and without anti-aliasing. We visualize the squared error at each pixel. As shown in Fig. A11, our anti-aliasing operation enables to reconstruct the original feature across the entire group accurately.
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Input Image
Downsampled features
Reconstruction
Reconstruction Error
Without Anti-aliasing
With Anti-aliasing (Ours)
Without Anti-aliasing
With Anti-aliasing (Ours)
Without Anti-aliasing
With Anti-aliasing (Ours)
Figure A11: Visualization of the reconstruction quality of the latent feature after subgroup subsam-
pling operation.
31


Reviews:
[{"review_id": "1OcA8zBXWm", "reviewer": "Reviewer_tNpv", "summary": "The paper proposes a framework for the subsampling of data in group equivariant architecture. The framework explains how to make sure the downsampling operations are anti-aliased, leading to an improved performance. Their approach shows how to choose a sub-sampled group to make sure that the map stays equivariant and how to make sure that data is not aliased.", "questions": "- Is this the \"first\" work on looking into the subgroup sampling theorem of groups? Xu et al. 2021 discuss group equivariant subsampling. How this work differs from this prior work? Are there other works in the literature that address the discussed issue? If yes, please include these in the intro.\n\n- It would be nice to show with a toy example why subsampling in groups is not an intuitive task of \"every other sample\" and what can go wrong. In this version of the paper, this intuition is not well-discussed.\n\n- Prior to this paper how the literature is doing sub-sampling on groups? Could Table 1 include such a comparison where other approaches fail?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper uses signal processing and group mathematical formulation to formulate how to properly downsample in group equivariance architectures. I appreciate the level of delicacy in discussing the network should be designed to \"properly\" process data to make sure that information is not corrupted by aliasing in each sub-sampling layer. The paper is written clearly and is well-organized.", "weaknesses": "While the method is thorough, the authors are losing an opportunity to demonstrate the implications and significance of anti-aliasing in an experiment. Given that accuracy and results are only shown on benchmark datasets such as MNIST and CIFAR, whether this approach would benefit large-scale experiments is unknown. Here are suggestions and clarifying comments for improvements:\n\nThe paper lacks information on how their work differs from the prior works listed in related works. For example, how is the literature currently performing such a task? Is the literature completely missing such a subsampling framework for equivariance networks, hence the absence of comparison baselines? Is the key message of the paper that prior works don't do anti-aliasing? Or they do sub-sampling differently, which results in a non-equivariant map.\n\nThe experiments can be more thorough. They are limited in its current stage.\n\n- Can the author focus on an application where the impact and benefits of anti-aliasing is better shown? Perhaps for an application where high-frequency information is of importance. An analysis on the artifacts and impacts of those due to naive sub-sampling can also help.\nResults on MNIST in its current form are limited.\n\n\nMinor\n\n- I suggest not using bold phrases (such as line 44) in the paper.\n\n- In Example 1, the middle label should be (b).\n\n- I suggest not coloring the Lemma, Claim, etc., and following ICLR formatting instructions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XDvDBR9Ewi", "reviewer": "Reviewer_fTfZ", "summary": "This paper proposes a method to subsample signals over groups by a given factor $R$. The authors, motivated, by signal processing, also provide an anti-aliasing operation to ensure bandlimited-ness and also provide a sampling theorem for groups.", "questions": "On top of the above weaknesses I had some further questions:\n\n- On line 57 the authors claim that the subsampling layer selects suitable subgroups for the tasks. There are a couple of issues with this: first of all, how is “suitable” defined in this context? In what way do the authors support that claim and how is it evaluated in their experiments? Finally, and maybe most importantly, the statement mentions tasks: however, their subsampling algorithm is task-agnostic and performs the subgroup choice without any knowledge about the downstream task. In what way, then, is the subgroup that is selected suitable for the task?\n- The definition of the minimal generating set in 125 is weird. The way things stand, since by construction $e\\notin S$, it can never hold that $\\langle S \\rangle = G$, for any $S$.\n- There are multiple typos in the footnote in line 323. There is also a typo in 343 (no space between $\\mathcal{M}$ and the previous word), and also on 501 $\\textrm{Acc}_{\\textrm{orbit}}$ appears twice.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper has strong connections to signal processing and the approach seems natural from the perspective of signal processing. Concepts are explained well and overall the paper is fairly well-written.", "weaknesses": "The paper has a few weaknesses, mainly:\n\n- unclear motivation,\n- unexplained choices.\n\n**Motivation**\n\nThere are two crucial choices that are made for the paper, but they are never discussed or motivated. Specifically:\n\n- Signals are defined over groups. This is highly non-standard in the equivariance literature, and it’s not clear how the methods presented here fit into the existing space of equivariant works. This is greatly illustrated in Section 5.1 and Figure 3, where the groups $C_{16}$ and $D_{16}$ are used essentially to label the nodes and have no effect on the actual signal. For rotationally-equivariant architectures, we fully expect the feature maps themselves to exhibit equivariance, which is not the case in this work. If that is not possible using this framework, this severely limits the applicability.\n- Interpolation in (8) is defined on a subsampled signal. This is highly canonical in signal processing, where interpolation can be applied independently of downsampling. It sees this is necessary for the authors’ analysis, however this is not motivated or explained at all.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VGrrfiNsZG", "reviewer": "Reviewer_FQDU", "summary": "This paper presents a method in the context of group equivariant neural networks, to design a subsampling operator where we can derive the subsampled group (called subgroup) and a corresponding anti-aliasing method.", "questions": "- In practice, is it always easy to obtain the minimal generating subset for a certain concrete group G? Or maybe another to phrase the question is: given a generating subset for a group, how easy is it to determine whether it is minimal?\n- In example 1, is $S = \\{\\delta t\\}$ ? and then we simply pick $s_d = \\Delta t$? \n- What does \"reconstructed perfectly\" mean in Claim 2?", "limitations": "", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- I really like that some examples are provided.\n- I think it's great to clearly illustrate what claims are tested and under which conditions in experiments.\n- Results show the interest of the proposed method in the given context.\n- Code is provided to allow the replication of the experiments. (I did not try running them)", "weaknesses": "My assessment is based on the lack of clarity of the paper. There small bits and pieces that are confusing but more generally concreteness and link to the application is lacking.\n\n- Clarity: As someone not familiar with group equivariant neural networks, it was very difficult for me to imagine what it means to have a signal indexed by elements of a group. I think section 4 would benefit from having a very visual example beforehand that can explain what all of the objects mentioned are in a concrete setting.\n- Experiments: it's not clear what exactly \"We apply the proposed subgroup selection and anti-aliasing operator to equivariant CNN architectures\" means. I don't understand what is implemented in the code unless I read it, if I stick to the paper. \n- Nit: when introducing the order of a generator, maybe it's worth specifying that it always exists and point to a reference.\n- Nit: \"the minimal generating subset\" -> it's not clarified beforehand whether there exists only one.\n- Nit (notations): \n. $V$ is at one point a vector space, then vertices on a graph. This is confusing. \n. The downsampling rate is sometimes $r$ sometimes $R$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J5pek9jRva", "reviewer": "Reviewer_rHa3", "summary": "This paper proposes a method for achieving equivariant (finite) group downsampling with anti-aliasing. The main contributions are twofold:\n1. Given a downsampling rate, the authors introduce an algorithm to identify an appropriate subgroup within the given finite group.\n2. The authors generalize the concept of a low-pass filter to signals over the group, enabling anti-aliasing and perfect reconstruction after downsampling.\n\nExperiments demonstrate that:\n1. The anti-aliasing low-pass filter achieves perfect reconstruction post-downsampling.\n2. Performance on image classification improves with rotated MNIST and CIFAR-10", "questions": "1. After subgroup downsampling, the model will only be equivariant to the subgroup. What is the rationale of imposing an equivariant constraint in Eq. 15 on the original larger group?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written, well-organized, and relatively easy to follow.\n2. The concept of automatically determining a subgroup for downsampling the original group signal, given a specific downsampling rate, is intriguing.\n3. The approach of identifying a smooth and equivariant low-pass filter (Eq. 15) before performing group downsampling is also interesting.", "weaknesses": "1. A main weakness (as pointed out by the authors themselves) are the limited experiments. The results reported are no where close to the readily-available benchmark on rotated MNIST [1].\n2. The authors argued that, in the MNIST experiment, the proposed equivariant-downsampling achieves comparable results with significantly smaller model size. However, one can argue that the original group C24 and D24 are too \"large\" and probably unnecessary to begin with. It would be interesting to see the comparison to a G-CNN without any subgroup downsampling and start with the smallest group used in the last layer of the proposed method. \n3. Also, the advantage of algrithm 1 (automatically determine a subgroup based on the subsampling rate) on Cn and Dn is not clear. I would be much interested to see its effectiveness when dealing with larger groups (such as permutation S(n)).\n4. Although I am no expert on the subject, but I would be surprised that perfect reconstruction for group signals after subsampling on a subgroup has not been considered in the literature. Can the authors cite more results on this subject and compare their result to the available ones?\n\n\n[1] Weiler, Maurice, and Gabriele Cesa. \"General e (2)-equivariant steerable cnns.\" Advances in neural information processing systems 32 (2019).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)