On the Fourier analysis in the SO(3) space : the EquiLoPO Network
Authors: Dmitrii Zhemchuzhnikov
Sergei Grudinin
Venue: ICLR 2025 Poster (2025)
Keywords: Equivariance
Fourier Analysis
SO(3)
MedMNIST3D
Local Activation
CNN
Group Convolution
Computer Vision
3D Medical Images
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We present 3D Group CNN with a Fourier representation in the SO(3) space and local activation in the Fourier domain

Abstract: Analyzing volumetric data with rotational invariance or equivariance is currently an active research topic. Existing deep-learning approaches utilize either group convolutional networks limited to discrete rotations or steerable convolutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivariance to Local Pattern Orientation on the continuous SO(3) group while allowing unconstrained trainable filters - EquiLoPO Network. Our key innovations are a group convolutional operation leveraging irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into a ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive evaluation on diverse 3D medical imaging datasets from MedMNIST3D demonstrates the effectiveness of our approach, which consistently outperforms state of the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation function, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at https://gricad-gitlab.univ-grenoble-alpes.fr/GruLab/ILPO/-/tree/main/EquiLoPO.

Published as a conference paper at ICLR 2025
ON THE FOURIER ANALYSIS IN THE SO(3) SPACE THE EQUILOPO NETWORK
Dmitrii Zhemchuzhnikoy!22 and Sergei Grudinin'
Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
AIRI
zhenchuzhnikkv@aaii.. sergei sergeee..rpaaaiieeee---reeeeeeee frr
ABSTRACT
Analyzing volumetric data with rotational invariance or equivariance is cur- rently an active research topic. Existing deep-learning approaches utilize ei- ther group convolutional networks limited to discrete rotations steerable con- volutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivari- ance to Local Pattern Orientation on the continuous SO(3) group while allow- ing unconstrained trainable filters EquiLoPO Network. Our key innovations are group convolutional operation leveraging irreducible representations as the Fourier basis and local activation function the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive eval- uation on diverse 3D medical imaging datasets from MedMNIST3D demon- strates the effectiveness of our approach, which consistently outperforms state the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation func- tion, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at / unt www....oo....................... grenoble- alees. fr/GruLab/ ILPO/-/tree/man Bquuiiooo//
INTRODUCTION
Deep-learning methods have shown remarkable success in analyzing spatial data across various do- mains. However, in many real-world scenarios, the data can be presented in arbitrary orientations. Therefore, the output of the neural network should be invariant or equivariant to rotations of the input. While data augmentation can partially address this requirement, it leads to increased compu- tational demand, especially for volumetric data, which have three rotation angles.
To tackle this challenge, researchers have developed equivariant neural network architectures that utilize rotationally equivariant operations. We can broadly categorize these methods into two classes: group convolutional networks and steerable convolutional networks. Group convolutional networks achieve rotational equivariance by convolving data in both translational and rotational spaces, but they are typically limited to discrete set of rotations. On the other hand, steerable convolutional networks employ filters that are analytically equivariant continuous rotations, but they impose constraints on the filter structures.
In this work, we propose novel equivariant neural network architecture that combines the strengths of both approaches. Our key contributions are:
1. We present group convolutional network that achieves analytical equivariance with spect to the continuous rotational space SO(3) by leveraging irreducible representations as the Fourier basis.
"The work for this paper performed while the author was at Univ. Grenoble Alpes, CNRS, Grenoble INP, IJJK; he is currently affiliated with ATRT.

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Published as a conference paper at ICLR 2025
2. Contrary steerable convolutional networks, approach does not impose constraints on the filter structures beyond the finite resolution limits.
3. We present a local activation function in the rotational space that provides a well-defined mapping from input function values to output function values, ensuring the preservation equivariance properties while avoiding the reduction of the architecture to a steerable convolutional network.
By addressing the limitations of existing methods, our approach offers a powerful and flexible frame- work for analyzing volumetric data in a rotationally equivariant manner without the need for data augmentation or constraints on filter structures.
2
RELATED WORK
2.1 DEEP LEARNING FOR VOLUMETRIC DATA AND EQUIVARIANCE
In recent years, deep learning has firmly established its place in the analysis of spatial data. Neural networks learn hierarchy of features by recognizing spatial patterns at various levels. However, in real-world scenarios, multidimensional data are often given in arbitrary orientation and shift, as the coordinate system is not defined. In such cases, it is desirable for the output of the neural network to be independent of the rotation or shift of the input data. While dealing with shifts is straightforward through the use of convolution, which is inherently shift-equivariant, achieving equivariance with respect to rotations requires additional efforts. A primary solution for achieving this effect is augmentation of the training dataset with rotated samples (Krizhevsky et al. 2012). Augmentation leads to an increase not only in the dataset size and consequently training time, but also in the number of network parameters needed memorize patterns in multiple orientations. In volumetric cases, this increased demand for computational resources can be overwhelming, as there are three angles of rotation 3D, compared to just one in 2D. For some data types, a canonical coordinate system may be defined (Pages et al. 2019, Jumper et al. 2021 Igashov et al., 2021, Zhemehuzhnikov et af.) (022). However, in most real-world scenarios, such a coordinate system cannot be identified, and even within a canonical coordinate system, the same local patterns may be encountered in different orientations. These circumstances have led the community to focus on methods that are analytically invariant equivariant to rotations, utilizing rotationally equivariant operations.
Rotational equivariant methods for regular data can be divided into two groups: group convolution networks and steerable convolutional networks. Our method formally belongs to the first group but without a specific approach to activation in the Fourier space can be reduced to method from the second group as it is shown in Appendix A Thus, we will briefly describe below group convolution networks, spherical harmonic networks and activation in the Fourier space.
2.2 GROUP CONVOLUTION NETWORKS
The pioneering method from the first class was the Group Equivariant Convolutional Networks (G-
CNNs) introduced by Cohen and Welling (2016a), who proposed general view convolutions in different group spaces. Many more methods were built up subsequently upon this approach (Wor-
rall and Brostow 2018 Winkels and Cohen 2018 Bekkers et al., 2018: Wang et al. 2019: Romero
et 2020 Dehmamy et al., 2021 Roth and Manaondld] 2021 Knigge et al. [022] Liu et al.,
2022b] Ruhe et al. 2023). Several implementations of Group Equivariant Networks were specifi-
cally adapted for regular volumetric data, e.g., BobbeNet(Worrall and Brostow 2018) and 3D G-CNN
[Winkels and Cohen 2018). Methods of this class achieve rotational equivariance by convolving data not only in translational but also in the rotational space. The authors of these methods consider a discrete set of 90-degree rotations, which exhaustively describe the possible positions of a cubic pattern on a regular grid. However, equivariance with respect to this discrete group of rotations does not guarantee equivariance on the continuous group SO(3). Separable SE(3)-equivariant network (Kuipers and Bekkers, 2023) approximates equivariance in SO(3) but lacks analytical equivariance since the authors sample only a finite set of points in the SO(3) space. Analytical rotational equiv- ariance in 3D can be achieved by using irreducible representations in the O(2) the SO(3) group.
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2.3 SPHERICAL HARMONICS NETWORKS
Cohen and Welling 2016b) introduced steerable networks, class of methods that use analytically equivariant filters with respect particular group of transformations. The first two approaches that applied aaalttialll-equiariinnn filters to the 3D data are the Tensor Field Networks (TFN)
(Thomas et al., 2018) and the N-Body Networks (NBNs) (Kondor 2018). [ondor et al. (2018) presented similar approach, the Clebsch-Gordan Nets, applied to data on a sphere. Weiler et al. (2018) presented steerable networks for the regular three-dimensional data. The authors deduced complete equivariant kernel basis where input and output irreducible features have arbitrary de- grees. The filter must belong to the subspace of the equivariant kernels. This requirement may limit the expressiveness of the network (Duval et al. 2023 Weiler et al.) 2023). To summarize, group convolutional networks do not put constraints on filters but provide equivariance only on a discrete space of rotations. Spherical harmonics networks use irreducible representations to achieve analyti- cal equivariance on the continuous rotational space but imply constraints on the filters. Our method employs irreducible SO(3) representations (Wigner matrices) the Fourier basis in a group convo- lutional network. This approach allows us to obtain analytical equivariance and avoid constraints on the filter shape. In general setup, such a convolution can be reduced in Appendix A because an SO(3) irrep can be seen as a set of O(2) irreps. However, we introduce a novel activation function in SO(3) that prevents the reduction of the whole architecture to steerable network. Indeed, our activation is not permutationally invariant with respect to Wigner coefficients of different orders and the same degree and thus cannot be reduced to a steerable network, as we demonstrate in Appendix A
steerable network, shown
2.4 ACTIVATION OPERATORS ON IRREDUCIBLE REPRESENTATIONS
Activation functions play a crucial role in neural network architectures as they introduce nonlinear operations. When working with irreducible representations (irreps) in onationally-equivarinn neu- ral networks, it is mandatory to choose activation functions that preserve the equivariance properties. Several approaches, listed below, have been proposed to apply activation functions irreps. Norm- based activation functions operate on the norm (magnitude) of each irrep, preserving the equivari- ance properties. The L2 norm of each irrep is computed, and a scalar activation function is applied to the norm. The activated norm is then used to scale the original irrep (Thomas et al. 2018). The same principle was used for the Fourier decomposition of a function in 3D by (Zhemchuzhnikov let al., 2022). Gated activation functions introduce learnable parameters to control the activation of each irrep (Weiler et al. 2018). A separate set of learnable weights is used to compute gating signal, which then applied to the irrep using element- wise multiplication. Capsule networks use special type of activation function called the squashing function Sabour et al. (2017). The squashing function scales the magnitude of the output vectors (irreps) to be between ( and 1 while preserving their direction. Tensor Product (TP) activation is a learnable activation function that operates on the tensor product of irreps (Kondor al., 2018). It applies learnable set of weights the tensor product of the input irreps and then projects the result back onto the original irrep basis.
The listed activation functions introduce non-linearity in equivariant neural networks. Unlike tra- ditional interpretations, we view irreducible representations (irreps) as Fourier coefficients of the SO(3) space, offering distinct perspective on their role in the network. From this viewpoint, we can classify activation functions as either global or local. Let fi and fout represent the input and output functions of an activation operation a. We call an activation global if
fout (fm)).
(1)
meaning that the value at any point in foor depends on the values at all points in fin Conversely, an activation is local if
faul (z) = a (ff(
(2)
for any point x. This implics that the value at any point of the output function depends only on the value the same point in the input function, establishing a direct and unambiguous mapping between input and output values in real space.
To the best of our knowledge, all the previously published activation methods in this domain global (Weiler and Cesa. 2019 Bekkers et al. 2024). This is suboptimal and can lead to noncompact representations and lack of feature hierarchy learning, which our approach aims to address. By
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integrating Wigner matrices within Fourier function framework in SO(3). method maps values unambiguously in real rotational space, aiming to approximate the ReLU operator in real space. We further support the significance of locality with computational experiments, showing that models with local mapping significantly outperform those with global mapping in terms of accuracy.
3
SE(3) CONVOLUTION
Patterns
B
Pattemn
*H
II
H
H
Invariant
Equivariant
pottorn
pattorn recognition
recogniiion
m
Figure I: Comparison of invariant and equivariant networks, subplots A and B, respectively. A layer with invariant pattern recognition cannot distinguish some images, producing identical feature maps for different inputs. In contrast, equivariant pattern recognition allows the discrimination of these images by building feature maps in 6D.
One can think of a six-dimensional (6D) roto-translational convolution that outputs three- dimensional (3D) feature map using orientational pooling (Zhemchuzhnikov and Grudinin 2024] Andrearezyk al. 2020 Kaba et al., 2023). This type of convolution is invariant to orientaionns of patterns in the input map. However, such a convolution approach is not the most expressive, as it cannot discriminate some input data. Figure [ schematically demonstrates toy example where a layer with invariant pattern recognition produces identical feature maps for two smiling and sad faces looking in different directions. potential solution would be maintain the output map in 6D: (3)...
ho R) = f/ (F) 1o).
(3)
gg
where fr) and w(r) are the input and the filter maps, respectively. This operation is equivariant to both orientations of f(r) and w(F") (for more details please see Appendix B). It is worth noticing that the equivariant property in the two cases above holds in different manners. both arguments F and R are rotated. In the second case, rotation R2 is applied only to the second argument. Besides, the expression in Eq. [T shows how to coordinate rotation of two components of arguments of 6D map. Introduction of such 6-dimensional feature map leads to the question how to treat such data. this paper, we present a novel equivariant convolution in 6D where both input and filter maps in SE(3) and the of rotations is continuous,
the first case,
[ dRo / drof(F Fo, Ro) R R-) S0(3)
1 R)
(4)
The usage of continuous space of rotations ensures analytical equivariance of the convolution. Below we will show how the irreducible representation helps to perform an integration in the SO(3) rotation space.
3.1 INTEGRATION THE SO(3) ROTATION SPACE
It is useful to express the rotational part of the convolution in the space of Wigner rotation matrices. Appendix C provides an essential theory on rotational expansions. Let us substitute expansion coefficients from Eq. C8 into the convolution operation 1tF,, R) in Eq. 4 and compute its expansion coefficients:
21, +1
1 (7)
dR
8T2
S013)
21, +
( dRo I dro. Fo, Ro)w(R Fo, R- RRo) (5)
dR D4
kika (R)
8 50(3)
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Let us now decompose functions f(r) and w(F) using Eq. C4 Eq. C2
and Eq.
C7
l
f(F Fo, Ro) Σ
(F Fo) D (Ro),
kaka
1= Ay
and
Lake
w(R R RRo)
Kg
Kig
w
1344 kyk
(r) (r)
ksko
(Ro) (R) (
where Lin and Lüter are the maximum expansion orders of the filter in the rotational and the three- dimensional Euclidean spaces, respectively. alsoo uses the unitarity property from Eq. C6 Eq.
S then simplifies to
Lin
lg
(7)= dro
pla kake (F+ ki kakka (Fo);
St
k1ka
Jza
1/
R.
where
8x
Liktr
ig K2 (ekkllkkkkkk))u (ro)
shila
kikakaka
(Ft)
Σ
21
4=0
Kig
Σ
(kkgllakzl4kg) (Drt))
(9)
Appendix D presents a more detailed derivation of Eq. Parameters w d2l4 kykkks (ro) are trainable. The output map expansion coefficients 11 (7") have maximum degree of Lou- We shall also stress that the values of Lin and Lom must satisfy the triangular inequality.
Lin - Lntterl I Lout S Lin + Liteer
(10)
The computational complexity of the convolution in Eq. 9 LIN Pae where N is the linear size of the input data, and Dm and Dout are numbers of input and output channels. Ap- pendix E proves the noto-translational equivariance with respect to the input data and the rotational equivariance with respect to the filter.
4 NON-LINEAR OPERATIONS
In neural networks, linear layers detect patterns and their probabilities, whereas nonlinear layers, specifically activations, selectively amplify these probabilities to form compact representations. We can classify activation functions as either global or local. Let fin and fout represent the input and output functions of an activation operation a. We call activation global if four (ffi), meaning that the value at any point in fou depends on the values at all points in fm- Conversely. (x)(())), for any point z. In our work, we aim to propose a local activation
activation
local if fa)) function for the Fourier representation of the data in the SO(3) space. Appendix [F] discusses in more detail why a local activation can useful. The equivariance of the local activation operator
proven in Appendix H
4.1 LOCAL ACTIVATION IN THE WIGNER SPACE
has been demonstrated that in NN architectures, on a certain interval, the ReLU function can be approximated polynomial of low degree (Ali et al. 2020 Leshno al. 1993 Gottemukkula
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2019).
Even the quadratic polynomial of degree two showed competitive results in the ResNet archi- tectures (Gottemukkula 2019). However, as mentioned in Appendix G the product of two functions defined in the SO(3) space with Wigner coefficients, has a higher resolution than each of the initial functions. Subsequently, to avoid the loss of information, we need to increase the maximum expan- sion order of the product, given the Wigner matrix expansion. For function in SO(3) defined with Wigner coefficients of the maximum degree L, the result of applying a polynomial of degree n Since the number of Wigner expansion coeffi- cients grows as the cube of the expansion order, to approximate the activation function, we chose the activation polynomial of the second degree. This paper studies multiple approximation strategies to this polynomial in the SO(3) space. Generally, the ReLU approximation expression can be written down as
will have the maximum expansion order of nL
(aa(F) D f(7) )
(11)
where P2(x:) co + ci 622² and D is scaling factor. Below, we discuss several approaches for this approximation in the Wigner space.
In the adaptive coefficients approach, approximate the ReLU operator [-3o(f) + (JJ) + n(O) interval, where (u, are the mean and the standard deviation of f(R), re- spectively,
11 [So(3) f(R)RR = f a=
[sors)( (S(R) - -2)
(fh, 21 +1
(12)
[so(3) dR
[Sor3) dR
ku 12=-1
There are three possible cases to consider. In the special case of +4 < 0, we make an assumption that f(R)- < 0, VR. Then, we put the polynomial approximation to y 0.01x, instead of y to avoid zero gradients. In the second special case of 3o + > 0, we assume f(R) > 0. VR and set y = z. In the general case, the function f(R) ranges both positive and negative values. For simplicity, we first divide the values of the function by 3o and then apply a polynomial that approximates ReL.U on [-1+k, + At). where k = # k E [-1, 1]. To find the optimal polynomial coefficients in this case, we formulate the following minimization problem,
min (4).
F(cu, C1, k),
(13)
where
dx(co + cræ + +ox++22222 c2 { 1+ dx(co + epæ + c20² x))². (14)
F(co,c1,02,h)
-14k
Then, we can consistently deduce (see details in Appendix
15 (1t*
2k + 1);
-/2 15k 26 33 +8;
3
(5 - 94: + +3k² 236.++70 +1). (15)
32
16
32
Figure I in Appendix [ shows the plot of these coefficients as function of normalized mean and the error of this approximation, respectively. Finally, to obtain the polynomial activation, we multiply the result by 30,
[(RR) 30
fac = 3o
(16)
In the second strategy with constant coefficients, we consider the denominator D equal to the third of the function norm [|/ In practice, this means that the function range lies in the [-1, interval. This is a special case of the interval from the previous approach at k = 0. Such a value of k leads to the following polynomial coefficient values,
15 : = -
3
15
C2 32
P2
+
x
+
2
(17)
32
32
2
32
We then multiply the result of the polynomial function by |l.f
f(R)
fac (11512/3))3
(18)
(ll[11//3)
We have also studied approach with trainable polynomial coefficients. Here, the denominator D is the same as in the previous case, D = |f However, the three polynomial coefficients (Co, C1 (2) trainable values.

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EquiLoPO ResNet Architecture
B
Sampled
Input (L=0, D=1)
Initial Block (L=2, D=4)
Basic Blook (L=2, D=4)
SoftMax
AugPool
Linear
Oulput
95% confidence interval
31,96 0.53X
SoftMax
Initial Block
2000
Convolution (L=2. D=2)
Activation (L=4, Dm4)
Convolution (L=1, Dn4)
Activation (L=2, D=4)
Basic Block
-2000
Canvolution
Activation (L=2, D=4)
Convolution D=4)
Activation (L=2, D=4)
-4000
(L=1, D=4)
-2500
2500
5000
Sampled maximum
Figure 2: A Schematic representation of the EquiLoPO ResNet- 18 architecture, with a sequence of operations in the Initial and Basic blocks. L is the maximum expansion degree of the last operator in the block, and D is the number of the block's features. B Correlation between the Sampled Maximum and its SoftMax approximation of the output functions in the rotational space in the trained model with locally adaptive coefficients on the Vessel dataset.
4.2 OTHER NON-LINEAR OPERATIONS
We incorporate additional non-linear operations essential for our framework, including SO(3) pool- ing, global activation functions, and normalization adapted rotational spaces. Detailed descrip- tions are provided in Appendix
5 RESULTS
Our method is centered applications involving regular volumetric data. Extending this method to irregular data would necessitate significant modifications that are beyond the scope of this paper. Consequently, we evaluated our method using collection of voxelized 3D image datasets. For this, we used MedMNIST v2, a vast MNIST-like collection of standardized biomedical images (Yang al., 2023). It is designed to support a variety of tasks, including binary and multi-class classification, and ordinal regression. The dataset encompasses six sets comprising total of 9, 9,998 3D images. All images are resized to 28 x8 x 28 voxels, each paired with its respective classification label. We used the 7 2).
split provided by the authors of the dataset (the proportion
We constructed multiple architectures with various activation methods, each reflecting the layer se- quence of ResNet-18 (He et al.) (016).. Figure A schematically illustrates our EquiLoPo model along with its main components. These include the initial block, the repetitive building block, pooling operators in SO(3) and 3D spaces, and a linear transformation at the end. Appendix K provides details these blocks. We also specifically adapted the Batch Normalization process for 6D (3D x S3(3) data.
Table 1 lists detailed comparison of our models' performance against the baseline models: various adaptations of ResNet (He et af.] 2016), featuring 2.5D/3D/ACS (Yang et al., 2021) convolutional layers, alongside open-source AutoML solutions such as auto-sklearn (Feurer al. 2019), and AutoKeras (Jin et al. 2019). We have also added results of the models that were tested on the collection: FPVT (Liu et 2022a), Moving Frame Net (Sangalli al. [2023). Regular SE(() con-
volution (Kuipers and Bekkers 2023) and ILPOResNet50 ((aeeeseee (ZZhemchuzhiikov and Grudinin 2024
They are also discussed in Appendix K
In MedMNIST, the classes are imbalanced, meaning that to measure the predictive precision of a model, we need other metrics besides accuracy (ACC). Consequently, we also consider the AUC- ROC (AUC) metric, which provides a more informative assessment of imbalanced datasets. Accord- ing to the metrics, we can observe that our models either outperform or demonstrate state-of-the-ar levels rounded to the third significant digit nearly all metrics, except for the accuracy on the organ dataset. The organ dataset is largely resolved, with all tested methods achieving a very high level of AUC. Additionally, the dataset might benefit from more channels (more than the four used) within the model's layers for improved performance. However, one of our goals was to limit the number of parameters in our model.

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Among the ofae-of-theear methods, only ILPOResNet (Zhemchuzhnikov and Grudinin 2024 has fewer parameters. The ILPOResNet model is based on the same principles of pattern definition and detection but carries recognition in an invariant manner while remaining in 3D. In contrast, the presented equivariant model offers a more expressive architecture, as evidenced by its performance on the collection datasets, but requires more parameters.
When considering the various types of activation we tested, three stand out with the best perfor- mance: local activation with trainable coefficients, local activation with adaptive coefficients, and global activation with trainable coefficients. Tables [L] L2 and L3 in Appendix [arrndixxsSitr show the results of experiments on architectures with these types of activations but smaller number of building blocks (I or 2 instead of 8). Local activation with adaptive coefficients demonstrates the highest robustness since, even in small models, performs relatively well.
Here, a reasonable question may arise whether the local activation is necessary if the global activa- tion already demonstrates comparable results. We shall notice, however, that global activation, the current implementation, employs SoftMax in the argument of the multiplier function, which in turn utilizes the ReLU approximation, form of local activation. In Appendix M we tested an alterna- tive strategy for global activation, replacing SoftMax with the 2-norm. As evidenced by the results presented in Table MI] this alternative strategy yields significantly inferior performance compared to the original approach with SoftMax.
Methods
# of prmus TIM
Organ
Nodule
Fracture
Adrenal
Vessel
Synapse
ACC
AUC
AUC
ACC
AUC
ACC
AUC
AUC
ACC
ResNer-18 (He et
2016 2021
0.977
0.788
0.838
0.835
0.387
0.451
0.718
0.772
0.748
0.846
0.634
0.096
2255DD Yang
I8
Lile
et
al.
33M
0.996
0.907
0.863
0.844
0,712
0.508
0.827
0,721
0.874
0.877
0.820
0.745
2016 +
3D
Yang al.
2021
ReiNet-18 18
He et
al
11M
0.994
0.900
0.873
0.847
0.714
0.497
0.839
0.754
0.9.0
0.928
0.708
0.722
Yang et
2021
Ko-5Not-50
[Hc
0.974
0.769
0.848
0.552
0.397
0.732
0.763
0.751
0.877
0.669
0.735
2016
2.5TX
Yang ot al.
2021
[e et
0.883
0.847
0.725
0.494
0.828
0.745
0.918
0.851
0.795
2016 +
3D Yang al
KeaNett-0
He et al
15M
0.994
0.889
0.886
0.841
0.750
0.517
0.828
0.758
0.912
0.858
0.719
0.709
2016
ACS Yang
2021
suld kleumm
Feurer al.
0.977
0.814
0.914
0.874
0.628
0.453
0.828
0.802
0.910
0.915
0.633
0.730
2019
AuroKeras
Dia
al.
0.979
0.804
0.844
0.834
0.642
0.458
0.804
0.705
0.773
0.894
0.538
0.724
TPP
Liu et al. 20222
0.923
0.800
0.814
0.822
0.640
0.438
0.801
0.704
0.770
0.888
0.530
0.712
(Sangalii
0.745
0.871
0.615
0.815
0.953
0.896
2023] al
Kegular SE(3) convolution and Bokkers)
172k
0.698
0.858
0.604
0.832
0.869
Kuupers
2023
1e.OResNNe.50
0.879
0.871
0.767
0.608
0.869
0.809
0.851
0.940
0,923
trainable activation
0.861
0.563
0.876
0.963
0.875
Local adaptive activation
418k
0.971
0.767
0.916
0.871
0.803
0.613
0.885
0.805
0.968
0.958
0.984
0.940
Local activation
418k
0.761
0.282
0.856
0.793
0.758
0.546
0.896
0.836
0. 0.805
0.890
0.883
0.847
Global trainable activation
113k
0.960
0.654
0.904
0.881
0.751
0.600
0.828
0.768
0.958
0.945
0.848
0.807
Global ulaptive activation
113k
0.735
0.180
0.674
0.784
0.567
0.438
0.745
0.785
0.680
0.887
0.671
0.727
Global constant activation
113k
0.754
0.203
0.730
0.794
0.620
0,500
0.843
0,768
0.672
0.885
0.584
0,287
Local trainable activation,
548k
0.944
0.598
0.739
0.794
0.624
0.379
0.507
0,768
0.709
0.885
0.620
0.345
const. resolution?
Table 1: Comparison of different methods on MedMNIST's 3D datasets. (*) For these methods, the
number of parameters is unknown. (?) Here, implemented the ResNet-50 architecture. In the other designs, we used the ResNet-18 versions.
5.1 ACTIVATION ANALYSIS
We previously highlighted the necessity of ensuring that the activation coefficients of the output possess a maximum degree higher than that of the input to prevent information loss when applying the polynomial, specifically, Lout 2Lin- However, investigating the scenario where high out- put degrees are discarded, aligning the maximum output degree with the maximum input degree ( Troat Tin) merits consideration. Figure illustrates the impact of the increased resolution on the
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Constant
Adaptive
0.
Bell Colse sayal
Bell
BiLl
Derpe serd
1:3
Oxtiral Sipud
Outpal Sipnall
1.3
-0.3
-10
Domitt
Deauity
Deaitt
Figure 3: Effect of the ReLU approximation with constant and adaptive polynomial coefficients in the SO(3) space for two resolution strategies: A,C low-resolution setup, Lout Lini B,D - -high- resolution setup, Lout = -iin- Plots A and B show results with constant activation coefficients. Plots Cnd D show results with adaptive activation coefficients. The z and y axes represent the input and output functions, respectively. The histograms above and to the right of the main plot display the distribution of these functions' values. The top histogram plots also compare the input distribution (blue) with the output distribution (red). Opacity of the main plots indicate the density distributions, overlaid with the ReLU functions.
approximation accuracy of the SO(3) ReLU function. For this plot, we analyzed one of the rotational distributions from the last convolution of the trained model with local adaptive coefficients the Vessel dataset reported in [ = 2) to observe how the activation operator, with constant (subplots A-B) and adaptive (subplots C-D) polynomial coefficients, operates under two conditions: Lout Lin and Lout Lin- We sampled 10t points in SO(3) and computed the values of both input and output functions at these points, presenting the results on a density plot.
Subplots A and C offer illustrations for the scenario with reduced resolution (Lour = Lin). Here, we note an inherent ambiguity between the values of the input and output signals. Indeed, each input function value maps to multiple output values. Such an inadequate mapping is also evidenced by the absence of sharp negative value cutoff. Subplots B and D showcase the polynomial ap- proximation with an enhanced output resolution. This setup eliminates the ambiguity in mapping in the real space, as each input function value uniquely corresponds to a single output function value. Moreover, this operator exhibits nonlinearity and closely approximates the ReLU function, depicted by a black dashed line. Consequently, negative input values transform into near- zero output values, phenomenon clearly visible in the right- histogram, which features prominent peak near y = 0 and a sharp cutoff of negative values.
The extent to which increasing the resolution enhances the performance of the architecture is evident in T] We conducted additional tests using a single architecture with a reduced resolution. This architecture employs local activation functions with trainable polynomial coefficients. As we can see from the table, the model with reduced resolution consistently underperforms across all datasets compared to the full-resolution model that also utilizes locally trainable coefficients.
In the case when the input function's mean is close to zero, the adaptive coefficients are almost identical to the constant ones and the two strategies have very similar effect as suggested by Eq. 17 In practice, zero-mean functions are frequent but there are also cases when the mean is shifted. Figure 3 demonstrates how the activation acts on a function with the mean significantly shifted to the negative values.
Subplots A and B demonstrate distribution of the input function defined in SO(3). The center of the distribution is clearly shifted to the left. In this case, the polynomial with adaptive coefficients (subplot D) approximates the ReLU function more precisely than the polynomial with constant coefficients (subplot B) because the approximation is closer to the ReLU function. Indeed, according to Eq. [4] the adaptive coefficients strategy gives twofold less approximation error than the constant coefficients one.
Table I presents the performance metrics for both strategies across the dataset collection. The choice between local and global activation significantly influences the performance. Therefore, our compar- ative analysis focuses more on contrasting local activations with adaptive versus constant polynomial coefficients. The strategy employing adaptive coefficients outperforms the one with constant coef-

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ficients all the datasets, except for Adrenal. This outcome is expected, adaptive coefficients typically provide a superior approximation of the activation function.
We evaluated global and local activation layers across three distinct strategies for the coefficients of the approximating activation polynomial. Table T the performance of global activation is comparable to that of the local activation. However, for both constant and adaptive coefficienss the local activation demonstrates superior results across all datasets.
indicates that for the trainable coefficients,
5.2 ANALYSIS OF THE SOFTMAX POOLING SO(3)
To examine the behavior of the SoftMax operation, we analyzed the outputs from the last convolution layer of the trained model with locally adaptive coefficients on the Vessel dataset reported in Table 1 We extracted the SO(3) function coefficients for all the maps, voxels, and channels, and then computed the SoftMax for these functions using Eq. J2 We sampled 10 points in the SO(3) space and reconstructed the functions at these points. Figure 2 demonstrates the relationship between SoftMax and the sampled maximum. is evident that there is high correlation between the two functions in the positive region of the sampled maximum.
Negative values in the sampled maximum lead to SoftMax values of zero. Near zero, uncertainty increases along with wide range of SoftMax values. This phenomenon appears due the poly- nomial approximation error. Specifically, the accuracy of the SoftMax approximation hinges on the condition that the mean of the output of the ReLU operation approximation must be greater equal to zero. Additionally, the mean of the SO(3) distribution, or its L1 norm, should be substantially smaller than the Ln-norm. While this condition holds in most instances, there are exceptions where it does not. These cases are clearly visible in Figure 2 as those providing high uncertainty in the SoftMax estimation near zero sampled maximum.
CONCLUSION
This work introduces a novel equivariant neural network architecture that achieves analytical rota- tional equivariance on the continuous SO(3) group while retaining the flexibility of unconstrained trainable filters. Our key innovations are group convolutional operation that leverages irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output function values in the real space.
Key Takeaways:
Enhanced Performance through Rotational Equivariance: By incorporating rotational equivariance, our models consistently outperformed or matched state-of-thh-an methods across various datasets in the MedMNIST collection. This underscores the importance of respecting rotational symmetries in volumetric data.
The use of higher- local activation functions with adaptive coefficients significantly improved the network's ability learn complex patterns, leading to better performance. This highlights the need for carefully designed activation functions in equivariant architectures.
Importance of Activation Function Design:
Effectiveness of SO(3) Pooling Operations: The SoftMax pooling operation in SO(3) proved effective in summarizing rotational features, which is essential for global and local prediction tasks.
Our findings emphasize the crucial role of rotation-equivariant operations and appropriate activation functions in deep learning models dealing with 3D data. Future work may explore extending these concepts irregular data structures integrating them with other forms of equivariance.
ACKNOWLEDGEMENTS
This work partially funded by MIAI@ Grenoble Alpes 1ANR-19-P31A-0003
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APPENDIX
A COMPARISON WITH STEERABLE NETWORKS AND IMPORTANCE ACTIVATION
As we have indicated above, the EquiLoPo convolution defined as
la
1 des
(F)
dr
plade Kigka
(f' Fo) EqquiLoP
bladddd (Fo), Kakaka
(A1)
K2k2
12 log
where
[SEquiLoPo
daaddd ta Ry Kakaka
(Fo)
8T²
Lyaet
PEquiLoPolke Iakaladadau (ro)
Σ
14
(4 (2 (R2))
212
+
Aig Ag==14
(A2)
and
12
(PEquiLoPo ky
dalg (ro)
(1 (2kilakis) (WEpuii)) Kigky
Illadd (ro).
(A3)
boy
daladiudue (ro) to be trainable but is equivalent In our implementation, we set coefficients [WiquiLoPo Rakyka 1h Hallldada (ro) is trainable because
to the case when [PEquiLaPol les ky
l+44
Σ
Σ
(1ikil2k7l4ks) |Fquui aPo ky
Halaladiada (ro) [WFquil oPo ko
dgl4dadeu (ro).
(A4)
1 1/2 k1==
Steerable Networks in the 3-dimensional case (Weiler et al. 2018 employ the following convolution operator:
12 plada (F Fo) Stterrablee
Iy (7)
dro
daad (Fo),
(A5)
=1=-4 l
where
[Ssterrablel kaka
paaadud (Fo)
Psteerable
(A6)
=
and coefficients [Pstecrable] dalalad (ro) are trainable.
If consider convolution in isolation from other operations in network, then our convolution operator is equivalent to the Steerable network convolution where the number of input and output
degree features are D = (21, + 1) and D' Dou (212 + 1) respectively:
[Psteerable] (2), +))+ ())))))))))) = [PEquiLOPo] d2 (ro).
1)+
(A7)
The difference between our method and Steerable Networks lies in the interpretation of the coef- ficients, namely we treat them expansion coefficients in SO(3). This interpretation results in special approach to activation. We propose such operation on the expansion coefficients that would correspond to the ReLU operator in the rotational space.
B EQUIVARIANCE OF THE CONVOLUTION WITH ROTATED FILTER IN 3D
Below we demonstrate equivariance of the 6D convolution to the orientations of the input f(7) map and the filter w(7) map. If f(F) - I(R5 F), then the convolution output
ho R) - /
[(R5 (R) (R)
13
f(R 1 Fo) w R
RoFo) ho ho, 1,, 1 R).
(B1)
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Consequently, if w(r) (Ri Fr,, then the convolution output
ho R) + { Fo)e 'R. - ho RoR).
(B2)
C THEORETICAL FRAMEWORK ON SPHERICAL HARMONICS, WIGNER, AND CLEBSCH-GORDAL COEFFICIENTS
This section provides overview of spherical harmonics, Wigner coefficients, and Clebsch-Gordan coefficients, essential mathematical tools in the field of quantum mechanics for the analysis of an- gular momentum.
C.1 SPHERICAL HARMONICS
Real spherical harmonics (I²( S + R are set of orthogonal functions defined on the surface of a sphere, denoted as 52 They are useful in expanding functions defined over the sphere and appear extensively in the solution of partial differential equations in spherical coordinates. The orthogonality of spherical harmonies expressed
J d (2) (Ω) ów
(CI)
82
where 5 stand for the Kronecker delta, signifying that spherical harmonics are orthogonal with re- spect to both the degree 1 and the order m. Any square-interrable function f(12) S - IR can be decomposed into real spherical harmonics as
f(l)) = ((Y
(C2)
1=0 K=-!
where
S = (())
C
are the spherical harmonic expansion coefficients. In practice, we use fixed maximum expansion order L defined by the resolution of input data.
C.2 WIGNER COEFFICIENTS
Wigner matrices, denoted as D D (R), form the irreducible representations of SO(3), the group of rotations in three-dimensional Euclidean space. These matrices rotation operations for spherical harmonics,
Y YA (R() = Σ D4y,, (R))
(C4)
1==-1
highlighting the transformation properties of spherical harmonies under rotation. The orthogonality of Wigner coefficients is held due to the following property,
J Sso13) dRD,, k2 (R) (R) = 21 + our Skgh4-
8n.
(C5)
Another property of a Wigner matrix is its unitarity,
121,, (R-1) = D',,, (R).
(C6)
Any square-integrabee function f(R) : S0(3) + can be decomposed into Wigner matrices as
((R) = Σ
Phine Dh.n,
(C7)
1=0 1o -1
where
/
(R)
[(R)D) (R)RR
(C8)
are Wigner matrix decomposition coefficients. Again, for practical considerations in Eq. C7 we will fixed maximum expansion order T.
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C.3 SEBSCH-GORDAL COEFFICIENTS
The Grrbsch-Gordan coefficients facilitate the coupling of two angular momenta in quantum me- chanics, leading composite states with well-defined total angular momentum. The interconnec- tion between Clebsch- Gordan coefficients, spherical harmonics, and Wigner matrices is illustrated through the integration of products of spherical harmonics and the transformation properties under rotation:
(21 1) + +1) (100O))))((KKKK 4r + +1)
(1)X (9)
(C9)
Sscta) dRD (R) (R)Dk (R)
8 2L +
(LK (LK,
(C10)
D FILTER EXPRESSION
In this section, we reveal how the expression in Eq. is obtained:
1g
Liter
24 +
(Loss) Kha (Ro) KORk (Ro)
shile
kikkksk4 (Fo)
8
1==0 ks
Aig
4=
La km=-14
(L dRD// ki
211 +
8m 8T2
(R. D/
Roks (R) (R)
w
ksk
Jala
(r) (S2r)
8T²
21 + 1 211 +
Ln
la
Laka
ōigis Okgko Bkake (hi Ma
Kokykk (r) (S2F)
1==0 Aos
la Ka=
1s Ry
1a
hig
La Ru
8T²
Lite
lg
(okeh (To))
(1a2ll2ks
Ials
Σ
(hhillkkz4kk))
) (S2F)
212 +
4=0
AAs==
(DI)
EQUIVARIANCE OF THE SE(3) CONVOLUTION
Here we prove the ato-traaslational equivariance of the convolution operator to the input data and the rotational equivariance to the filter.
E. EQUIVARIANCE WITH RESPECT TO THE INPUT FUNCTION
Let [(,, be rotated by R1. Then consider the output of the convolution in Eq.
[ dRo / dio (Ro Ri To, RT Ro) - R RRo)
pa
= -1
dRo
di (R Fo. Ri R ( R R RR))
SS(()
=f dRo drof 1
F Ro)w(R RiFo, R (R) 'iRoo) = 'R,,RFFFRR F, RR) (E1)
SS0(3)
R3
E.2 EOLTVARIANCE WITH RESPECT TO THE FILTER FUNCTION
Let w(F, R) be rotated by R1. Then consider the output of the convolution in Eq.
[ dRo / di Fo. Ro) ( 'R Fo. R -R''R-"
Ro) (R,')))
hF R,
(E2)
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F THE NEED FOR LOCAL ACTIVATION FUNCTION
In neural networks, linear operations typically alternate with nonlinear ones; the latter are often referred to activations. Nonlinear operations are necessary approximate complex dependen- cies in the data. However, from the perspective of learning hierarchies of patterns, the two types of operations have different purposes. Indeed, linear layers are tasked with detecting patterns and fetching a quantity proportional to the probability of finding a certain pattern. Nonlinear layers, in turn, penalize low pattern probabilities, allowing only high ones to pass and often serve learn compact representations. An incorrectly chosen activation can lead to noise accumulation when passing through multiple layers and also suboptimal latent representations. A classical activation solution is ReLU and its variants, which, despite some drawbacks, are the most intuitively under- standable activation functions. The vanilla ReLU operator passes only positive values of its input. In convolutional and other networks dealing with spatial data, the critical property that differentiates activation operations from the rest of the network is their locality, i.e., the activation is applied to different points in the Euclidean space independently.
In our case, as we have already mentioned, the output of the linear convolution is six-dimensional: R SO(3). Since the RR3 dimension discretizes into voxels, we apply the activation each voxel separately. The most interesting question, however, is how to design the activation function for the SO(3) data dimension. We define the distribution of values in SO(3) in our architecture with Wigner matrix decomposition coefficients. Each coefficient, similarly to the Fourier series. represents the global properties of the entire SO(3) distribution rather than an individual point in SO(3). However, we aim to design a local activation for SO(3) in the space of Wigner coefficients.
previously used approach (Zhemchuzhnikov and Grudinin 2024 Cohen al., 2018) consists in sampling the SO(3) space and requires a transformation from the Wigner representation into the ro- tation space, an activation in the SO(3) space, and an inverse transformation into the Wigner space. However, this approach loses the analytical SO(3) equivariance. It may not be a problem since sampling a sufficiently large number of points in the rotation SO(3) space can guarantee effective equivariance. However, this strategy inevitably leads to increased computational complexity. Conversely, the proposed Wigner coefficient representation, allows using few values to define a function in SO(3) and retaining analytical equivariance at the same time. There expressions for the decomposition coefficients of (eI.U(((R)) given the spherical harmonics coef- ficients of (R). However, we will use the analytical expression of the coefficients of product of two functions, given in Eq. HS in Appendix G It enables us to find an analytical expression for the spherical harmonics coefficients for a polynomial applied function SO(3).
analytical
G PRODUCT OF TWO FUNCTIONS IN SO(3)
The Wigner matrix decomposition coefficients of a product of two functions defined in SO(3) can be expressed through coefficients of two multiplier functions.
Let functions fi (R) and (2(R) have coefficients [fi]k and [filk and (proa(R) be the product of (R) and (2()):
Speod (R)){(().
(GI)
Then
its [Smaalle kyke
(laks (1aks/hiklaka) (Takee (2laaaa [fal
Ra=-12
(G2)
where [frood are coefficients of fpood RR), L1 and L2 maximal degrees of f1 and fa respec- tively. In order to avoid loss of information the maximal degree of fpoo is L3 = L + L2. Thus, the product operation requires increase of resolution of data.
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H EQUIVARIANCE OF THE LOCAL ACTIVATION OPERATOR
We define our local activation operator by
faalF) = P3 f(r)
(HI)
with
P2(x) = + c1 C222
(H2)
Thus,
(r)(() c e f
02 PM)
(H3)
D
The equivariance of the constant term D and the linear term (f immediate. The nontrivial part is show that the quadratic term (F) is also equivariant.
Let us express the function f(r) with its Wigner expansion.
f() Σ
(H4)
I=0 k2=
so that its square (i.e., product with itself) is expressed via
[ ks
Σ
(Ia ki l K1, la ka) (1z ko l K2, ka) Mi Uik
1200 k1
Ry lig
(H5)
Under a rotation R E SO(3), the Wigner coefficients transform as
[fodle
Σ D'mna (R) lfl'may. m=-
(H6)
Substituting these rotated coefficients into (H5) and using the properties of the Clebsch-Gordan coefficients (which couple the representations) along with the unitarity of the D matrices, one obtains
ls Ao
Σ
D (g) 1
(H7)
m=-ls Lg
In other words, the squared function transforms
(()) which is exactly the equivariance property. Therefore, the entire local activation operator,
(H8)
fma(F) coD f() + - (
(H9)
satisfies
fra(F) - fac(R 'T) proving its equivariance.
(H10)
I DERIVATION OF EXPRESSIONS FOR ADAPTIVE COEFFICIENTS
The solution of Eq. 14 satisfies the following conditions,
0F/0co 4ep 4c k + 4c2 (k2 + ()) (k + 1)² = 0
0F/0c 4cnk 4er (k2 ()) + (k3 + kk [k 113
(11)
0F/0cz = 4co(k² + ( + 4c (k* k) + 4c2(k+ + 24 + (k: =
Multiplying the first equation by and (k2 + 1/3) and subtracting it from the second and third equations, respectively, we get
G + 2c0k = - (k3 - 3k 2) c + (k2 + 1
(12)
(3k+ 10k 8k +
Figure shows the plot of these coefficients as a function normalized mean k and the error of this approximation, respectively.
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0.005
0.8
0.004
0.6
Value
Error
0.003
0.002
0.2
0.0
0.000
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
Figure 11: Left: Coefficients of the polynomial as functions of 30 Right: Approximation error as
function of J
30
J OTHER NON-LINEAL OPERATIONS
J.1 MAX POOLING OPERATION IN THE CONTINUOUS SO(3) SPACE
For wide range of tasks, whether they involve global or local (voxel-wise) prediction, is necessary to design an operation that reduces the representation of the function in the rotation space to a single value. For brevity, average pooling of function in the SO(3) space, entails discarding Wigner coefficients of all degrees higher than zero. However, we aim design an operation analogous to max pooling.
may call it pooling operation in SO(3). The most straightforward approach,
Let us note that the polynomial approximation from Eq. Hlloow for the simulation of the SoftMax effect in SO(3). More precisely, we can define SoftMax as
SoftMax(f) = SS(3) w (R)))
(JI)
where weight w(R)- = ReLU(f(R)) [so(s) (R')(((R')ddR' The weight function estimates the ratio of (R)) the positive part of the function ((R). Thus, can also define the SoftMax function in SO(3) using only the Wigner matrix expansion coefficients of ((R) as
SoftMaxpot (f) = SSora) act act(f(R))
I15a 12 Ilfa 11
8 Evs My 2F ((a))
(J2)
Tso(3) act(f(R))dR
[fa
We should note that there can be different approaches to SoftMax simulation, of polynomial coefficients. In scenarios where the pooling operation follows the ReLU operation, there is no need apply the activation function again. Instead, we directly calculate the ratio of the 2-norm squared of the input function to the pooling operation to its 1-norm.
with the usage
J.2 GLOBAL ACTIVATION
Above, we introduced local activation function that requires higher resolution and, consequently, an increased number of coefficients. This raises the question of whether the performance improve- ment offered by this operation justifies the associated rise in computational cost. As a baseline for the activation, explored operation where the output resolution is identical to that of the input. The formalism presented above allows us to define the global activation,
SGlobAct R (lobAct([(R))) = a (WooftMax(f)) + ())
(J3)
where we use SoftMax from Eq. j2 apply o = sigmoid, and then multiply the result by the function fRR). W and b are trainable parameters. Linearity of the Wigner matrix decomposition gives us the following expression for the Wigner coefficients of the global activation function,
[Jelonnckk ( (WSoffMaax()) b)l[k,
(J4)
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This expression can be seen gating mechanism that only passes 50(3)-distributiin with suf- ficiently high trainable positive values, at least for a single point in SO(3). Although the input and output resolutions of this operation are identical, still employ a higher resolution (expansion order) in the ReLU approximation within the SoftMax operation.
J.3 NORMALIZATION
The fact that the mean and the standard deviation of function in the rotational space can be ex- pressed in terms of Wigner coefficients allows us to introduce an expression for the normalization of function in SO(3):
(R) ((R) - +
(J5)
where i and a are the mean and the standard deviation of function f. respectively, and y and are trainable coefficients. We can also extend this operation for data in R33 x SO(3), where the Fuclidean component discretized and characterized by a set of functions fizK (R), where i, j, and & are voxel indices. The batch normalization is then defined follows:
[foxkln = [ija(()
Hiyk + +3,
(J6)
Jisk
where Pajk and Jijk are the mean and the standard deviation of a function fiyk. respectively, and N Eigh Pijk and 0² = N Eighh Lig.. with N being the number of voxels.
K DATASETS, ARCHITECTURES AND TECHNICAL DETAILS
Our method is centered applications involving regular volumetric data. Extending this method to irregular data would necessitate significant modifications that are beyond the scope of this paper. Consequently, we evaluated our method using a collection of voxelized 3D image datasets.
K.1 DATASETS
We assessed our designs on MedMNIST v2, vast MNIST-like collection of standardized biomed- ical images (Yang al. 2023). It is designed to support variety of tasks, including binary and multi-class classification, and ordinal regression. The dataset encompasses six sets comprising a to- tal of 9,998 3DD images. All images are resized 28 28 28 voxels, each paired with its respective classification label. We used the train-validation-tex split provided by the authors of the dataset (the proportion is 7: 1 2).
K.2 BASELINE ARCHITECTURES
For our baselines, we employed the same model configurations that the creators of the dataset uti- lized for testing on MedMNIST3D datasets datasets(Yang al. (003). These include various adaptations of ResNet (He et al. 2016), featuring 2.5D/3D/ACS Yang et al. 2021 convolutional layers, alongside open-source AutoML solutions such auto-sklearn (Feurer et al. | 2019). and AutoKeras (Jin al. 2019). We have also added results of the models that were tested on the collection: FPVT (Liu al. 2022a), Moving Frame Net (Sangalli al. (2023). Regular SE(3) convolution (Kuipers and Bekkers (023)) and 1ee50 ((Zhemchuzhnikov and Grudinin, 2024h). Table tested architectures.
Jists all the
K.3 TRAINED MODELS
We developed multiple architectures with various activation methods, each reflecting the layer se- quence of R8 (He et al.) 2016). The final architecture, featuring reduced-resolltion activa- tion. is inspired by ResNet-50 (He et al. 2016). Figure KT schematically illustrates our EquiLoPo architecture along with its main components. These include the initial block, the repetitive building block, pooling operators in SO(3) and 3D spaces, and a linear transformation at the end. Table K3 details the basic block for SE(3) data in ResNet-18. The initial convolutional block, outlined in
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Table K2 initiates the architecture. We also specifically adapted the Batch Normalization process for 6D (3D x SO(3)) data.
It is important architecture), which increases activation resolution, results in an eightfold increase in the maximum degree. This surge is attributed to triple consecutive activations without intervening convolu- tion, potentially escalating computational demands. Consequently, our exploration was limited to the ResNet-18 architecture, where the activation operation enhances output resolution. Figure K1 illustrates the EquiLoPO ResNet-18 architecture.
that the Bottleneck block in ResNet-50 (the simplest building block of the
scheme -2..pdf
EquiLoPO ResNet Architecture
Input (L=0, D=1)
Initial Block D=4)
Basic Block
(L=2.
SoftMax
AvgPool
Linear
Output
x8
Initial Block
Basic Block
Convolution
Aclivation (L=4, D=4)
Convolution (L=1, D=4)
Aclivation (L=2, D=4)
Convolution
Activation
Convolution
Activation
(L=2, D=2)
(L=1, D=4)
(L=2, D=4)
(L=1, D=4)
(L=2, D=4)
Figure KI: Schematic representation of the EquiLoPO ResNet-18 architecture, with a sequence of operations in the Initial and Basic blocks. L is the maximum expansion degree of the last operator in the block and the number of the block's features.
Every filter in our trained networks is confined within 3 parameterized by weights wfal kakkks (r:), where T; 0, 1, /2, v represents all possible radii from the center in the cubic space. We used the ADAM optimizer (Kingma and Ba, 2014). In order to avoid overfitting, we apply the dropout operation after each activation operation. The learning rate of the optimizer and the dropout rate are hyperparameters. All the models are trained for 100 epochs. Table lists hyperparameters optimized for validation data performance. T44 presents memory and time consumption metrics for EquiLoPOResNet
volume in 3D. Filters
ILPOResNet-15 and ResNet-18 18 models in the inference mode. The current implementation consumes significantly more memory than the vanilla ResNet architecture and requires longer execution time because the activation is coded in Python rather than C++. Transitioning the implementation to C++ could significantly reduce both memory and CPU footprints.
Organ
Nodule
Synapse
activatinn Local adapore
0.01
0.01,
0,0055
0,01,
constant activation
dr-
0:000,
activatonn
00-00
adopiive sctivation
0:0
10:0000,
mainable activation,
0.01
0.01,
0.01
1r-0001. 0.01
Ir- 0.01, 0.01
0,01,
cons. resolution... (*)
Table K1: Optimal hyperparameters for the trained networks: learning rate (Ir) and dropout rate (dr) of the trained networks. (1 Here, we implemented the ResNet-50 architecture. In the other designs, we used the ResNet-18 18 versions.
Step
Operation
Details (size of the filter in 3D; output, in- put and filter maximum degrees)
Convolution, Eq.
3x3x3, Lin = 0, Lout = 2, Lilter 2
Batch Normalization, Eq.
J6
Lm 2, Loui 2
3
Local Activation, Eq.
H
Lm = 2, Loui =4
Convolution, Eq.
8
3x3x3, Lin 4, Lout 1, Latter
5 Batch Normalization, Eq.
J6
Lin 1, Lout =
Local Activation, Eq.
Tin 1, Tomn 2
Table K2: Sequence of operations the Initial convolutional block of Net
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Step
Operation
Details (size of the filter 3D; output, in- put and filter maximum degrees)
Convolution, Eq.
3x3x3, Lin = 2, Lout 1, Liiter =
Batch Normalization,
Eq.
J6
Lin 1, Loat
Local Activation, Eq
HT
Lin = 1, Lou = 2
Convolution, Eq.
3x3x3, Lin 2, Lout
1, Leiter
5
Batch Normalization, Eq.
J6
Lin = 1, Lout =
Addition
Add imput (1 0, 1) to the output
7
Local Activation, Eq.
H1
Li Loui = 2
Table K3: Sequence of operations in the Basic block of EquiLoPOResNet-18
Model
Memory, GB
GFLOPs
Inference
time per batch of 32 samples, seconds
ResNet-18 18
2.47
35.52
0.03
ILPOONe-18
7.14
19.58
0.3
EquiLoPO
27.02
68.44
2.49
(local activa-
tion)
EquiLoPO (global acti-
30.05
53.75
2.13
vation)
Table K4: Memory, FLOPs and Time consumption for EquiLoPOReesNtt118,, ILPOResNet-18 and ResNet-18 inference.
L PERFORMANCE WITH FEWER BUILDING BLOCKS
In the main experiments, we evaluated the proposed neural network architecture using the standard ResNet- configuration with 8 building blocks. To further investigate the method's performance and scalability, we conducted additional experiments by reducing the number of building blocks to and 2, respectively, for the three activation functions that demonstrated the best performance the main experiments. These were local activation with trainable coefficients(Tablll LI local activation with adaptive coefficientsTTable L2). and global activation with trainable coefficients L3).
As the number of building blocks decreases from 8 to 2 and then to 1, there is general trend of degrading the performance across all datasets and activation functions. The extent of performance degradation varies across datasets. For example, the OrganMNIST3D dataset exhibits more sig- nificant drop in accuracy (ACC) when reducing the number of blocks compared to other datasets like NoduleMNNSS33D VesseIMNIST3D..
The three activation functions (local activation with trainable coefficients, local activation with adap- tive coefficients, and global activation with trainable coefficients) show different levels of robustness to the reduction in building blocks. Local activation with adaptive coefficients (Table 5) maintains relatively high performance even with fewer blocks, compared the other two activation functions.
Methods
of
Organ
Nodule
Fracture
Adrenal
Vessel
Synapse
prms
AUC
ACC
AUC
ACC
ACC
AUC
ACC
8 Blocks
418k
0.991
D.866
0.923
0.861
0.727
0.563
0.876
0.792
0.950
0.958
01965
0.878
I Block
1366
0.936
0.489
0.872
0.861
0.726
0.508
0.885
0.846
0.963
0.914
0.876
0.861
Table LI: Performance comparison for ResNet-like architecture with 1, 2 and 8 blocks with local activation and trainable coefficients.
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Published as a conference paper at ICLR 2025
Methods
of
Organ
Nodule
Fracture
Adrenal
Vessel
Synapse
prms
ACC
ACC
8 Blocks
418k
0.977
0.767
0.916
0.871
0.803
0.613
0.885
0.805
0.968
0.958
01.084
0.940
Blocks
176k
0.970
0.929
0.871
0.725
0.885
0.832
0,970
0.935
0.346
0,806
Block
13kk
0.961
0.659
0.914
0.852
0.786
0.596
0.909
0.862
0.963
0.932
0.920
0.824
Table L2: Performance comparison for ResNet-like architecture with 1, 2 and 8 blocks with local activation and adaptive coefficients.
Methods
Organ
Nodule
Fracture
Adrenal
Vessel
Synapse
AUC
ACC
AUC
ACC
AUC
ACC
AUC
ACC
AUC
ACC
AUC
ACC
8 Blocks
113k
0.900
0.854
0.904
0.881
0.751
0.000
0.828
0.768
0.958
0.945
0.848
0.807
Blocks
43k
0.960
0.643
0.894
0.858
0.746
0.533
0.887
0.829
0,929
0.816
0.810
1 Block
32k
0.939
0.546
0.894
0.852
0.722
0.538
0.875
0.842
0.923
0.911
0.841
0.813
Table L3: Performance comparison for ResNet-like architecture with 1, 2 and 8 blocks with global activation and trainable coefficients.
M ALTERNATIVE STRATEGY FOR THE GLOBAL ACTIVATION
In this section, we test alternative strategy for global activation. Instead of using Eq. J3 we use
falobAc( (R) = GlobAct(f(R)) = +) ())(
(MI)
where we replace the SoftMax function with the 2-norm of the function. Table MI compares the performance of the ResNet-18-like architecture with global activation using trainable coefficients and the two different strategies: SoftMax and 2-norm. The results show that using the SoftMax function consistently outperforms the 2-norm alternative across all datasets.
Methods
of AUC
Organ
Nodule
Fracture
Adrenal
Vessel AUC
Synapse
ACC
ACC
AUC
ACC
AUC
ACC
AUC
ACC
SoftMax 2-norm
113k
0.960
0.654
0.904
0.881
0.751
0.600
0.828
0.768
0.958
0.945
0.848
0,807
0.733
0.220
0.584
0.794
0.602
0.383
0.711
0.768
0.609
0.539
0.730
Table MI: Performance comparison for the ResNet- ---Net-----ike architecture with global activation using trainable coefficients and different aggregation functions.
23


Reviews:
[{"review_id": "xBhWAbWViT", "reviewer": "Reviewer_HZTy", "summary": "The paper describes a class of equivariant group convolution layers based on the steerability paradigm (Wigner-D irreducible representations). Working with such continuious representations of SO(3) signals requires working with feature fields of Fourier coefficients, and one cannot simply take arbitrary activation functions. Hence the authors also propose a new class of activation functions based on polynomial approximations of ReLU, which maintain equivariance exactly. This is an important contribution. Next they show the method works well on 3D volumentric data (Medical MNIST), outperform several state-of-the-art equivariant baselines.", "questions": "see above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Being able to build efficient and effective equivariant architectures for 3D volumetric data is still an active field in geometric deep learning. The paper provides a valuable contribution, primarily for the following\n- The proposed activation functions are a great contribution, because it is well known that simple norm-based activations (like gating and the \"global 2-norm\" based one of appendix L) do not work very well. The authors show that the proposed activation function is much more effective.\n- The method is sound and well-presented, and the related work section is complete (however, it is mostly absent from the main body as it is in the appendix).", "weaknesses": "## Weaknesses\n- Context is missing about where the methods stands relative to the equivariant deep learning literature. Some references should be included in the main body.\n- The equivariant convolution layer itself is, in my opinion, not a new contribution as it falls in the steerable class of g-cnns. \n- In the experimental section, an important comparisons for the activation functions is missing, namely comparing to inverse Fourier based activation functions [Fourier o ReLU o InverseFourier]. This is often used to by-pass the constraint of norm-based activations in Fourier space.\n\n## Detailed Comments\n\n### On Group Convolutional Networks Classification\nThe introduction briefly introduces group convolutional networks into two classes: group conv nets and steerable conv nets. But in my opinion they are both the same as steerable convs (tensor field type) are identical to regular g-convs when expanding the conv kernels in a basis of irreps. So a better separation would be regular vs steerable group convolutions. The first discretizes the group and directly implements equation 1 and 2, the second adopts a fiber bundle viewpoint and talks of steerable feature fields (features that transform via irreps). The kernel constraint becomes more intricate in the latter case, but has the advantage that you maintain exact equivarance (up to the spatial discretization) as you don't need to work with grids over SO(3).\n\nI would say the present work falls into this category, and you have implicitly solved the kernel constraint by deriving the convolution kernels via a Fourier transform from the regular group conv formulation (an approach that is also taking in this lecture [link](https://youtu.be/EBzqL1OXigM?feature=shared) when deriving harmonic networks from regular group convolutions. These two papers make a point of this connection in their appendices:\n\n- Brandstetter, J., Hesselink, R., van der Pol, E., Bekkers, E. J., & Welling, M. Geometric and Physical Quantities improve E(3) Equivariant Message Passing. In International Conference on Learning Representations (2021).\n- Bekkers, E. J., Vadgama, S., Hesselink, R., Van der Linden, P. A., & Romero, D. W. Fast, Expressive $\\mathrm{SE}(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space. In The Twelfth International Conference on Learning Representations (2024)\n\n### Technical Questions and Suggestions\n- When mentioning canonicalization, do you mean works like [kaba et al 2023]? Either way, please add cites when discussing it as an option.\n- When talking about 3D -> 6D -> 3D (invariant) convolution you cite Zhemchuzhnikov and Grudinin, which is very appropriate, but please also consider including [Andrearczyk and Depeursigne]:\n\nrefs:\n- Kaba, S. O., Mondal, A. K., Zhang, Y., Bengio, Y., & Ravanbakhsh, S. (2023, July). Equivariance with learned canonicalization functions. In International Conference on Machine Learning (pp. 15546-15566). PMLR.\n- Vincent Andrearczyk, Julien Fageot, Valentin Oreiller, Xavier Montet, Adrien Depeursinge Proceedings of The 2nd International Conference on Medical Imaging with Deep Learning, PMLR 102:15-26, 2019.\n\n### Equation Concerns\nCould you double check the equations for group convolutions. I have the impression that in Eq. 1 it should read $...w(R^{-1} r_0) …$ instead of $...w(R r_0)$. Same for Equation 2 which I think should read $...w(R^{-1}R_0, R^{-1} R_0)$, here with $R^{-1}$ multiplying on the left of $R$. I could be wrong here but I think there's an inconsistency between group correlation vs convolution where the translation is in convolution form, and rotation in correlation form.\n\nAlso in Eq 1 the integration measure is put at the end of the integral, and in 2 they are given up front.\n\n### On Section 3.1\nIs my understanding of the derivation of these steerable convolutions—as described above about regular vs steerable g-convs—correct? If so, how does this justify the claim that your method \"does not impose constraints on the filter\", since having to parametrize the kernels in a basis of wigner-D basis is in a way designing the kernel to automatically satisfy the constraint. But it would then still fall into the class of steerable layers, as described in full generality in:\n\nCesa, G., Lang, L., & Weiler, M. (2022). A program to build E(N)-equivariant steerable CNNs. In International conference on learning representations.\n\n### On Activation Functions\nI like the comment about the need for local activation functions. This could motivate why we typically see that regular group convolutions are more effective than steerable once, because they localize information *and* are compatible with powerful actiation functions like ReLU. This is the main motivation behind works s.a. [Bekkers et al. 2024, see ref above] and extensively tested for in 2D in:\n\nWeiler, M., & Cesa, G. (2019). General e(2)-equivariant steerable cnns. Advances in neural information processing systems, 32.\n\nSections 4.1 and 4.2 are nice contributions.\n\n### On Experimental Section\nI would highly encourage testing these activation functions also in 3D point cloud settings. There should be various benchmarks with models based on e3nn that could serve as a baseline. \n\nPlease improve the experimental sections with explicit conclusions and take-home messages. There's quite a lot to parse.", "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": "4lHZ7FVa2P", "reviewer": "Reviewer_tXtN", "summary": "The paper presents a deep learning network that guarantees rotational equivariance over continuous SO(3). It relies on a Fourier representation of volumetric data. The proposed model follows the well-established ResNet architecture where activations, pooling and normalization layers are adapted to the proposed data representation.", "questions": "1. Is it possible to compare the degree of rotational equivariance attained by the model with that attained by alternative methods?.\n\n2. If evaluated on a single seed, is it possible to report mean and standard deviation on both ACC and AUC-ROC?\n\n3. Results in Table 1 shows improved results depending on the activations used in the model. Still, it is unclear how to select the model in real applications, particularly since not in all cases the proposed method outperforms alternative methods. How can an end user choose the best model given an arbitrary dataset?\n\n4. How does the 6-D feature map affect memory consumption and training/inference time? Is it possible to report these numbers and compare them to alternative methods?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "The paper presents an in-depth analytical explanation of the proposed data representation and the components used in the proposed residual model. It also includes detailed analysis of components such as softmax as well as constant and adaptive nonlinear activations.", "weaknesses": "The main paper heavily relies on the appendix for several of the explanations, experimental settings and architectural details. Additionally, the introduction lacks context and references to better understand the motivation and current limitations of state-of-the-art algorithms. Authors should consider adding examples of previous work on group equivariance and the use of steerable filters in SO(3) in the main paper in an organized manner.\n\nFigure 1 is not very informative about the proposed method. For instance, a more detailed diagram might be useful to show the audience how the feature maps in 6D are constructed.\n\nI found the explanation in Section 3 to be hard to follow since:\n- h is used to express multiple convolution operations.\n- The main paper heavily relies on equations in the appendix to introduce formulation.\n- Several equations are contrived and do not comply with the paper's format (e.g. Eq. 5).\n\nThe analysis of the proposed layers in the main paper is limited to the activation functions. It may be interesting to ablate most if not all of the components proposed by the paper (normalization, pooling, conv. parameters, etc.) and how each affects accuracy, ROC curves and rotation equivariance.\n\nEvaluations are limited to accuracy and AUC-ROC. The degree of equivariance and how it compares to alternative methods is not evaluated.\n\nIt is unclear how to select the architecture to obtain the best performance. Table 1 reports 7 different configurations, each having a significantly different performance across datasets Additionally, Table 1 does not report if the metrics were computed on multiple seeds or a single one, so it is not possible to fully assess which technique is the best and in what case.\n\nThere are several grammar and typographical errors (lines 34, 280, 284).\n\nI think the paper requires both additional experiments and a manuscript revision. The way it is currently organized, it is hard to distinguish between the proposed ideas and those of previous work. The experimental section also lacks an ablation study. Since it relies on a single table and no rotational consistency metrics, it is not clear how each component affects the final classification performance.", "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": "RyQiqCoGev", "reviewer": "Reviewer_h7TR", "summary": "This work presents an equivariant neural network architecture named EquiLoPO Network, and innovates the design of Fourier basis and a local activation function in SO(3) space. Experiments on various databases along with the comparison with various models are given.", "questions": "1.\tPlease enrich the related work section, and clarify the motivation.  \n2.\tPlease investigate the latest literature and give comparison. \n3.\tPlease give more experiments details mentioned in the above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThis paper proposes a novel equivariant neural network architecture that combines the strengths of group convolutional networks and steerable convolutional networks by leveraging irreducible representations as the Fourier basis.\n2.\tThis paper presents a local activation function in the rotational space that provides a well-defined mapping from input function values to output function values.\n3.\tThe theoretical derivation is detailed, solid and reasonable.\n4.\tQualitative and quantitative experiments demonstrate the advantages of the proposed method on the diverse MedMNIST3D collection of 3D medical imaging datasets, and ablation studies verify the effectiveness of the proposed independent module. \n5.\tThe presentation is fluent, easy to follow, and understandable, with fewer grammatical and expression errors.", "weaknesses": "1.\tThe introduction does not detail the research background and related literature, and the research motivation is not sufficient; it only mentions the advantages of combining the two. \n2.\tThere is a lack of comparison with the latest literature. \n3.\tThe detailed computational cost of different ResNet-style architectures should be given, including training time, number of model parameters, time and space complexity.", "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": "7BESCHFJ16", "reviewer": "Reviewer_BUBE", "summary": "The paper introduces an SO(3) equivariant operator, where filters are unconstrained. The authors introduce various pointwise operations, such as polynomial nonlinearities and pooling that are placed in spherical harmonic space. The main innovation is how the polynomial nonlinearities are constructed, which does not introduce extra higher order harmonics into the spectrum, which would otherwise break a Nyquist-like property. They call their new model the EquiLoPO Architecture. They test there system on regularly-sample 3D volumetric medical imaging data, in particular the MedMNIST v2 dataset.", "questions": "Line 34: I would argue that the “increase computational demand” of 3D rotational data augmentation is negligible to the cost of training and running a neural network. Indeed equivariance is useful for its guarantee of symmetry and increased sample efficiency. Indeed, moving to 6D convolutions, with complexity in line 148, would be far more expensive from a computations standpoint.\nLine 38: The distinction between group convolutional and steerable networks is somewhat arbitrary. Both classes are types of group convolutional network. The major difference between them is the choice of group representation (regular representation vs. irreducible representation). I would change the naming here to avoid confusion.\nWhy are there no citations in the Related Work section? Perhaps this section should just be end of the introduction?\nLine 90: Please elucidate what you mean by orientation pooling. A diagram or equation would suffice.\nLine 94: Why take a direct product of R^3 and SO(3) when SE(3) has a semi-direct product structure?\nSection 3.1: I would suggest introducing notation in the body of the paper rather than requiring readers to dip into the Supplement. It is quite hard to follow, unless you already know the area fairly well.\nSection 3.1: How is this exposition different from the typical SE(3)-equivariant convolutions with a regular representation on the spatial component and using “steerable” convolutions on the rotational component? Is the derivative needed here?\nLine 155: How do you arrive at the statement that linear layers detect probabilities and that activations “selectively amplify these probabilities to form compact representations”. Do you have a citation for this?\nLine 157: I think people also use the term “pointwise” as well as local.\nTable 1: What measures do you take to make sure that architectures are comparable? They all have vastly differing numbers of parameters. Do you match on, say, FLOPs? Otherwise it is hard to compare numbers on a like-for-like basis.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "**Clarity**\n\nThe written quality of the paper is good. It is clear at each step what the authors intend.\n\n**Quality**\n\nI read all the math in the main text and as far as I could tell it was correct.\n\n**Originality**\n\nI believe the design of novel pointwise nonlinearities that limit the introduction of higher order harmonics into the spectrum of hidden representations is novel. It is certainly an interesting area.\n\n**Significance**\n\nI believe the introduced nonlinearity could be very interesting to the equivariance community. Good nonlinearities that can be applied in the harmonic space have long been sought after, yet none have been found to suffice. For instance “General E(2)-Equivariant Steerable CNNs” _Weiler & Cesa_ (2019) explores some variants, but found that performing nonlinear operations in the physical domain to be most effective (most likely because locality is maintained). Having to transform back to the physical domain is expensive and an Achilles’ Heel to that class of methods. In this sense, the nonlinearities introduced here are significant.", "weaknesses": "**Clarity**\n\nI found that the exposition of the paper makes it hard to understand, unless the reader is very acquainted with the area of equivariance. My suggestion to the authors is to streamline the math, improve notation, explain notation/variables in the main text, and explain what the math means in practical terms to the reader.\n\n**Quality**\n\nI have no doubts about the mathematical quality of the paper, that is sound. My major doubt concerns the experiments. It is not clear to me how to judge Table 1 without extra information, which I hope the authors will provide. I would like to know how I should compare models in an apples-to-apples fashion. Are they FLOP-adjusted? Otherwise, improvements of one method over the other cannot be compared. Furthermore, are non-equivariant baselines trained with data augmentation? That would only be fair.\n\nMaybe I have missed it (please correct me if I have), but as far as I can see there are no ablations that corroborate the assertions made that the introduced polynomial activations functions are superior to standard ones (ReLU GeLU, Swish, etc). It would have been useful for the paper to have an ablation of that sort, to strengthen the claims of the authors.", "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": "L3tWNmlFht", "reviewer": "Reviewer_DFKW", "summary": "The paper presents a novel equivariant neural network, the **EquiLoPO Network**, designed to achieve analytical equivariance on the continuous SO(3) group. The model utilizes **group convolutional operations** based on irreducible Fourier representations and introduces a local activation function to improve the flexibility and rotational equivariance of filters. Extensive evaluation on 3D datasets, particularly the MedMNIST3D, demonstrates that this approach outperforms existing methods in handling volumetric data.", "questions": "1. **Computational Efficiency and Practical Applications**:\n   - Need the complexity analysis of the computations required for continuous SO(3) convolution, how does the model perform on standard hardware compared to more traditional 3D convolutional networks? \n   - Considering the existence of the current foundation model, like DINOV2, that provides general patterns that works well for multiple tasks, is the new convolution still needed?\n\n2. **Generalization to real clinic Data**:\n   - The model is tested on 28×28×28 voxel dataset. Has the EquiLoPO network been tested on datasets outside of the resized MedMNIST3D collection? Especially the high-dimensional clinic data.  What would be the observations, particularly regarding performance on datasets where the continuous rotational symmetry might not be as critical?\n\n3. **Activation Function Details**:\n   - Could the authors elaborate on the decision to use local activation functions with trainable polynomial coefficients in the SO(3) space? In particular, what was the rationale for choosing the polynomial degree used, and were there alternative activation functions considered?\n   - Are there specific types of data for which the adaptive coefficient approach might be more suitable than the constant coefficients?\n\n4. **Impact of Batch Normalization on 6D Data**:\n   - Batch normalization for 6D (3D x SO(3)) data is mentioned as a key feature. Could the authors share more about the challenges or considerations involved in adapting batch normalization for this high-dimensional data? Were there any unique issues encountered when tuning these parameters?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This work represents a significant advance in multiple areas:\n1. **Fundamental**: The introduction of a continuous SO(3) equivariant convolutional model offers a mathematically grounded framework for analyzing volumetric data with rotational invariance.\n2. **Methodological**: The combination of group convolutional operations and a novel local activation function is a notable innovation that enhances the robustness and versatility of SO(3)-based neural networks.\n3. **Technological**: Implementing this model within a ResNet structure for practical 3D medical imaging tasks positions this work as an innovative tool for advancing medical imaging analysis.\n   \nThis work’s capability for true rotational equivariance without discrete approximation or filter constraints represents a new methodological benchmark in volumetric data analysis. The findings have broad implications for medical imaging and potentially other domains where the rotational orientation of data impacts model performance. By reducing the need for data augmentation and enabling models to operate more efficiently on 3D data, this work could significantly enhance model performance in fields requiring 3D data processing, such as autonomous navigation and robotics.", "weaknesses": "- The model’s computational complexity could be a limitation for deployment in resource-constrained environments. Further analysis of model efficiency in practical scenarios would be beneficial.\n - All images are resized to 28×28×28 voxels, which makes this like a toy example. How well does the model generalize to real medical datasets, and are there specific aspects of the activation function that might limit its applicability to other 3D datasets? \n - Clarity on the scaling factor in Equation 9 and its impact on activation function accuracy would improve reproducibility.\n - Details on the limitations of the SO(3) continuous convolution relative to discrete methods, such as potential downsides in specific edge cases.\n - Clearer explanation of how model parameters were tuned for the MedMNIST3D dataset, especially given class imbalances.\n - Additional insights into the batch normalization adaptations for 6D data would be useful for implementation clarity.", "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)