Incorporating Arbitrary Matrix Group Equivariance into KANs
Authors: Lexiang Hu
Yisen Wang
Zhouchen Lin
Venue: ICML 2025 poster (2025)
Keywords: equivariant neural network
Kolmogorov-Arnold network
Lie theory
Primary area: deep_learning
TL;DR: The first attempt to combine equivariance with KANs

Abstract: Kolmogorov-Arnold Networks (KANs) have seen great success in scientific domains thanks to spline activation functions, becoming an alternative to Multi-Layer Perceptrons (MLPs). However, spline functions may not respect symmetry in tasks, which is crucial prior knowledge in machine learning. In this paper, we propose Equivariant Kolmogorov-Arnold Networks (EKAN), a method for incorporating arbitrary matrix group equivariance into KANs, aiming to broaden their applicability to more fields. We first construct gated spline basis functions, which form the EKAN layer together with equivariant linear weights, and then define a lift layer to align the input space of EKAN with the feature space of the dataset, thereby building the entire EKAN architecture. Compared with baseline models, EKAN achieves higher accuracy with smaller datasets or fewer parameters on symmetry-related tasks, such as particle scattering and the three-body problem, often reducing test MSE by several orders of magnitude. Even in non-symbolic formula scenarios, such as top quark tagging with three jet constituents, EKAN achieves comparable results with state-of-the-art equivariant architectures using fewer than $40\\%$ of the parameters, while KANs do not outperform MLPs as expected. Code and data are available at [https://github.com/hulx2002/EKAN](https://github.com/hulx2002/EKAN).

Incorporating Arbitrary Matrix Group Equivariance into KANs
Lexiang Hu
Yisen Wang 12 Zhouchen Lin 123
Abstract
1991; Braun & Griebel, 2009), which offers an alternative Multi-Layer Perceptrons (MLPs) (Haykin, 1998; Cybenko, 1989; Hornik et al., 1989). Unlike MLPs, which utilize fixed activation functions on nodes, KANs employ learnable activation functions on edges, replacing the linear weight parameters entirely with univariate functions parameterized as splines (De Boor, 1978). On the other hand, each layer of KANs can be viewed spline basis functions followed by a linear layer (Dhiman, 2024). This architecture allows KANs to achieve better accuracy in symbolic formula represen- tation tasks compared with MLPs, particularly in function fitting and scientific applications. Subsequent works based on KANs have demonstrated superior performance in other areas, such as sequential data (Vaca-Rubio et al., 2024; Genet & Inzirillo, 2024b;a; Xu et al., 2024), graph data (Bresson al., 2024; De Carlo et al., 2024; Kiamari et al., 2024; Zhang & Zhang, 2024), image data (Cheon, 2024b;a; Azam & Akhtar, 2024; Li et al., 2024a; Seydi, 2024; Bodner et al., 2024), and so on.
Kolnogorov-Aroold Networks (KANs) have seen great success in scientific domains thanks to spline activation functions, becoming an alterna- tive to Multi-Layer Perceptrons (MLPs). How- ever, spline functions may not respect symme- try in tasks, which is crucial prior knowledge in machine learning. In this paper, we pro- pose Equivariant Kolmogorov-Arnol Networks (EKAN), a method for incorporating arbitrary ma- trix group equivariance into KANs, aiming to broaden their applicability to more fields. We first construct gated spline basis functions, which form the EKAN layer together with equivari- ant linear weights, and then define a lift layer to align the input space of EKAN with the fea- ture space of the dataset, thereby building the entire EKAN architecture. Compared with base- line models, EKAN achieves higher accuracy with smaller datasets fewer parameters on symmetry-rclatee tasks, such particle scatter- ing and the three-body problem, often reducing test MSE by several orders of magnitude. Even in non-symbolic formula scenarios, such as top quark tagging with three jet constituents, EKAN achieves comparable results with state-of-the- -rr equivariant architectures using fewer than 40% of the parameters, while KANs do not outperform MLPs as expected. Code and data are available https//ggthhbb.comm/uxxX///////
However, KANs themselves perform poorly on non- symbolic formula representation tasks (Yu et al., 2024). One reason for this is that splines struggle to respect data type and symmetry, both of which play important roles in ma- chine learning. Many recent works utilize symmetry in data to design network architectures, achieving better efficiency and generalization on specific tasks. For example, Con- volutional Neural Networks (CNNs) (LeCun et al., 1989) and Group equivariant Convolutional Neural Networks (GC- NNs) (Cohen & Welling, 2016) leverage translational and rotational symmetries in image data, while DeepSets (Za- heer et al., 2017) and equivariant graph networks (Maron et al., 2019; Keriven & Peyré, 2019; Satorras et al., 2021) ex- ploit the permutation symmetry in set and graph data. Equiv- ariant Multi-Layer Perceptrons (EMLP) (Finzi et al., 2021) propose a general method that allows MLPs to be equiv- ariant with respect to arbitrary matrix groups for specific data types, thereby unifying the aforementioned specialized network architectures.
1. Introduction
Kolmogoroy-Aroold Networks (KANs) (Liu et al., 2024b;a) are a novel type of neural network inspired by the Kolmogorov-Aroold representation theorem (Tikhomirov,
'State Key Lab of General AI, School of Intelligence Sci- ence and Technology. Peking University "Institute for Artificial Intelligence, Peking University 'Pazhou Laboratory (Huangpu), Guangzhou, Guangdong, China. Correspondence Zhouchen Lin <zlin @ dalin@pku.cot pww.ccdu.cn'..
Inspired by these equivariant architectures, we propose Equivariant Kolmogonov-Arool Networks (EKAN), which embed matrix group equivariance into KANs. By specifying the data type and symmetry, EKAN can serve as a general framework for applying KANs to various areas. Section introduce the preliminary knowledge of group theory. In
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Incorporating Arbitrary Matrix Group Equivariance into KANS
Section 3, we summarize related works. In Section 4, we construct layer of EKAN. We add gate scalars to the input and output space of each layer, and define gated spline basis functions between the input and post- aestvvaciittion space. To ensure equivariance when linearly combining gated basis functions, we construct the equivariant constraint and solve for the equivariant linear weights. In Section 5, we build the entire EKAN architecture. We insert lift layer before the first layer and discard the gate scalars from the output of the final layer, so that the input and output space of EKAN can be consistent with the original dataset. In Section 6, we evaluate EKAN on tasks with known symmetries. We show that EKAN can achieve higher accuracy than baseline models with smaller datasets or fewer parameters. Sec- tion 7, we conclude this work. In Figure I, we compare the architectures of KANS and EKAN.
2. Background
Before presenting related works and our method, we first introduce some preliminary knowledge of group theory.
Groups and generators. The matrix group G is sub- group of the general linear group GL(n), which consists of n x n invertible matrices. Each group element g E G can be decomposed into a continuous and finite component g 9192- We can obtain the continuous component 91 from a Lie algebra element A E g through the exponential A map exp: : g + G. i... 91 = exp(AA = Ek=n RT. Repre- senting the space where the Lie algebra resides as basis (A.) we have 91 = exp (E 04 A; On the other hand, the finite component 92 can be generated by a set M of group elements { and their inverses h- h formally speaking 92 = II hR. Overall, we can express the matrix group element as:
N
Kolmogorov-Araak Network (KAN)
N II
g = exp Σ a
h
(I)
Equivariant Kolmogorov-Amold Network (EKAN)
where {A:)e1 are called infinitesimal generators and [ i= are called discrete generators. We introduce com- mon matrix groups and their generators in Appendix A.
Group representations. The group representation PV G - GL(m) maps group elements to m x m invert- ible matrices, which describes how group elements act on the vector space V = RT through linear transformations. For 91.92 E G it satisties (g(g)))) (v Similarly, the Lie algebra representation is defined as dpv 9 - gl(m), and for A1, A2 E g. we have (Av(A) + A2) = dpv + (Ay( We can relate the Lie group representation to the Lie algebra representa- tion through the exponential map. Specifically, for A e g. pv (cxp(A)) = exp(dpr (A)) holds. Then, combining with Equation (I), the matrix group representation can be written as:
Figure 1. Comparison of the architectures of Kolmogorov-Amold Networks (KANs) and Equivariant Kolmagorov- Arnold Networks (EKAN).
In summary, our contributions are follows: (1) We pro- pose EKAN, an architecture that makes KANs equivariant to matrix groups. To our knowledge, EKAN is the first attempt to combine equivariance with KANs, and we expect that it can serve as a general framework to broaden the ap- plicability of KANs to more areas. (2) We specify the space structures of the EKAN Layer and define gated spline basis functions. We theoretically prove that gated basis functions can ensure equivariance between the gated input space and the post-activation space. Then, we insert a lift layer to pre- process the raw input feature, which aligns the input space of EKAN with the feature space of the dataset. (3) Exper- iments on tasks with matrix group equivariance, such as particle scattering and the body problem. demonstrate that EKAN often significantly outperforms baseline models, even with smaller datasets fewer parameters. In the task of non-symbolic formula representation, where KANs are not proficient, such as top quark tagging with three jet con- stituents, EKAN can still achieve comparable results with oftt--of-hheann art equivariant architectures while using fewer than 40% of the parameters.
pv(g) exp Σ a.ddy (Aid
II pv (ba.).
(2)
We can construct the complex vector space from the base vector space using the dual (+), direct sum (#), and tensor product (8) operations. To give a concrete example, let Vi and 1/2 be base vector spaces. The multi-channel vec- tor space, matrix space and parameter space of the linear mapping V12 + 1/2 can be represented as V1 @ V2, Vi & V2, and V22 & Vr. respectively. In general, given a matrix group G. we can normalize a vector space U into a polynomial- like form with respect to the base vector space V of G (the space where the group representation is the identity map- ping pv(g) (g) = g: intuitively, the transformation matrix
2

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Incorporating Arbitrary Matrix Group Equivariance into KANS
the matrix group element itself):
(Wang et al., 2020; Dangovski al., 2022; Lee et al., 2022; Garrido et al., 2023; Gupta al., 2024) has also achieved outstanding results. Based on these theoretical frameworks, equivariant networks are widely applied in various fields, such as mathematics (Zhao et al., 2023), physics (Wang et al., 2021; Hu et al., 2025), biochemistry (Bekkers et al., 2018; Winkels & Cohen, 2019; Graham et al., 2020), and others. EMLP (Finzi et al., 2021) embeds matrix group equivariance into MLPs layerwise, which we discuss in detail in Appendix C.
U +
T(pa (a)
+ VPn @ (V*)
where VPa
VOVO... and (V*)
Pa
& V* Its group representation and Lie alge-
Ra bra representation can be generated by the following rules:
(v.(()) = (g TT,
Kolmogorov-Arnold Networks (KANs). KANs (Liu et al., 2024b;a) place learnable activation functions on the edges and then sum them to obtain the output nodes, replac- ing the fixed activation functions applied to the output nodes of linear layers in MLPs. Formally, the 1-th KAN layer can be expressed as:
Pvivs (g) PV (g) pv)(g);
Pvi (g) = PV, (g) @ PV) dpv- (A) = (A),
(4)
dpvievs (A) = devi (A) + dpv: (A). dpvsovs (A) = dov, (A) En dpv, (A):
21+1, Σ dISu()),)) j 1,....,, M1+1, (6)
where @ is the direct sum, 69 is the Kronecker product, and is the Kronecker sum. We provide concrete examples of the space structure Appendix help readers understand it intuitively.
where is the number of nodes in the I-th layer, Ii, is the value of the i-th node in the 1-th layer, and dija is the activation function that connects Iι, to I4+1 In practice, Øt.j. consists of a spline function and silu function. The spline basis functions are determined by grids, which are updated based on the input samples. Then we can write the post-activation of Φt. as (11,) where G, k, and Br,, represent the number of grid intervals, the order, and the b-th basis function of splines node rı,j respec- tively. Therefore, the KAN layer be viewed spline basis functions B1,b and a silu function, followed by linear layer with Wijj.i parameters (Dhiman, 2024);
Equivariance and invariance. The symmetry can be di- vided into equivariance and invariance, meaning that when transformation is applied to the input space, the output space either transforms in the same way or remains unchanged. Formally, given a group G, a function f U, + U, is equivariant if:
Σ G-k-1 wi..s. BL.,,, ( Htw.g.....ssllu.........
bi e G po(g) f()())))))
(5)
where Pi and Po are group representations of U, and U.. respectively. Specifically, when po(g) = I. the function f is invariant.
R2
G4k-1 Σ
=
3. Related Works
+ +w...c++sil((muu) j= -....,, R441. (7)
Equivariant networks. Equivariant networks have gained significant attention in recent years due to their ability to respect and leverage symmetries in data. GCNNs (Cohen & Welling, 2016) embed discrete group equivariance into tradi- tional CNNs through group convolutions. Steerable CNNs (Cohen & Welling, 2017) introduce steerable filters, which provide a more flexible and efficient way to achieve equiv- ariance compared with GCNNs. Subsequently, SFCNNs (Weiler et al.. 2018b) and E(2)-equivariant steerable CNNs (Weiler& & Cesa, 2019) extend GCNNs and steerable CNNs to continuous group equivariance, while 3D Steerable CNNs (Weiler et al., 2018a) extend these models to 3D volumetric data. On the other hand, some works use partial differential operators (PDOs) to construct equivariant networks (Shen et al., 2020; 2021; 2022; He et al., 2022; Li et al., 2024b; 2025). Furthermore, equivariant self-suuer learning
4. EKAN Layer
In this section, we construct the EKAN layer, which is equivariant with respect to the matrix group G. First, define the space structures and explain their relationships. Then, we introduce gated basis functions and equivariant linear weights, which together form a layer of EKAN We summarize the space structures and network architecture of the EKAN layer in Figure 2.
4.1. Space Structures
A key aspect of equivariant networks is how group elements act on the feature space. Therefore, unlike conventional
3

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Incorporating Arbitrary Matrix Group Equivariance into KANS
actual output feature
T3
gale
output space U.
galed output space U
equivariant linear weights
post-aciivation space Une
specified
Do
Bo
B
gated basis functions
B
input space U,
gated input space Ui
Tp
Ti
control flow
actual input feature
Figure 2. (Left) The space structures of the EKAN layer and their relationships. (Right) The architecture of the EKAN layer, which consists of gated basis functions and equivariant linear weights.
networks, which only focus on the dimensions of the fea- ture space, equivariant networks need to further clarify the structure of the feature space. For example, for the group SO()). two feature spaces U1= V + R² R² and U2 = V @ V = R2 @ R have different group representa- tions pV, and Pt., but conventional networks treat them as the same space U = R-.
We summarize the aforementioned space structures and their relationships in Figure 2 (Left). The user first specifies the input space U₃ and the output space U, for the EKAN layer. Then the gated input space Ugi and the post-activation space Un are calculated based on U,, and the gated output space Ugo is calculated based on U., The actual input feature in Ugi passes through gated basis functions to obtain the acti- vation value in Un. which then passes through equivariant linear weights to obtain the actual output feature in Ugo-
We specify the input space and the output space of EKAN layer as U2 and Uo. respectively. Their structures can be normalized into the form of Equation (3). In particular, for ease of later discussion, we extract the scalar space terms To = (0, 0) and rewrite them as:
4.2. Gated Basis Functions
Since spline basis functions are nonlinear and have relatively complex iterative forms, directly solving the equivariance constraint in Equation (5) quite challenging. Recently, gar- ing mechanisms have been widely used in various areas, not only in language models to improve performance (Dauphin et al., 2017; Shazeer, 2020; Gu & Dao, 2023) but also in the design of equivariant networks (Weiler et al., 201 8a; Finzi et al., 2021). Inspired by this, we propose gated basis functions to make the basis functions in KANs (spline basis functions and the silu function) equivariant. Suppose that the input feature dgi E Uga can be decomposed according to the space structure shown in Equation (9):
U. = cT @
A T(P pia)
a=l
Uo coTo +
+ Ao a=1
T(Po,a: go,a)
where Pi o1 Po.a. ao Cs2 Co. A, Ao E N, Ps.a + n.a > O, Po.a + ao.a > O, and cTp To + To 1 ...
+ To-
We have to emphasize that the actual input/output feature does not lie within U,UU,. To align with gated basis func- tions, we add a gate scalar To to each acalsc term T(pia: (PPo,a ( in U./U, to obtain the actual in- put/output space. The mechanism behind this approach will be discussed in detail in Section 4.2. We denote this actual input/output space as the gated input/output space U Ugo
Ugg
- a=1
$i
+
A a=1
Vi.a
+
S/ya a==1
(10)
Ugi = To (
+ -==
T(Pia, (haa)
① A,To,
where S E To and Via E T(ps.a, Ri.)). For the non-scalar term Viaa, we apply the basis functions to the corresponding gate scalar s¹,a and then multiply the result by Us For the scalar term si,a, we consider it as its own gate scalar, which is equivalent to applying basis functions element-wise 8ia- Formalizing the above content, the post-activation value U. Um can be written as:
Ugo co +
+ Ao DA=, T(po.a )o + A.TO.
As shown in Equation (7), we split the KAN layer into basis functions and linear weights. From this perspective, we correspondingly construct gated basis functions and equiv- ariant linear weights to form the EKAN layer. We refer to the space where the activation values reside after gated basis functions and before equivariant linear weights as the post-activation space Un- The structure of Um depends on U, which we will elaborate on in Section 4.2.
G+k
Um
θ h
Um, by
(11)
4

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Incorporating Arbitrary Matrix Group Equivariance into KANS
where
The coefficients of each term in {um,bl0=0 1 are equal:
8i,0 (
(B(
+
D 1==1
Vi,a Bo((
Vg E G t0,1,..., k+: W = (G).
b << G k.
(18)
Um,bb
Si
silu
+
A
Ut (()
Flattening the linear weight blocks (Wil0=0 1+* into vectors, we obtain:
= G + k.
(12)
Ag G. (0, Pgo. (g) vo)(W)),
G
Note that 6 E C,To fore, we obtain the structure of the anst-activation space Un:
+ A,
a( (.a)) = U. There-
(19)
where Pgo.i Pgo @ p is the group representation of Uga @ U.. We use the same method as EMLP (Finzi et al., 2021) to solve for the equivariant basis Q and equivariant projector P = QQ of vec(W)). Similar to the transition from Equation (27) to Equation (28) (in Appendix C), de- compose the group representation Pgo.: into discrete and infinitesimal generators to obtain the following constraint:
Un (G
(13)
The following theorem guarantees the equivariance between the gated input space and the post-activation space (see Appendix D for proof).
Theorem 4.1. Given matrix group G, the gated input space Ugi and the post-activation space Um can be ex- pressed in the forms of Equations (9) and (13), respectively. The function f Ugi - Um is defined by Equations (10) to (12), that is, Vm = f( Then, f is equivariant:
vb E {0, 1,....... G k)
dp (A1)
Vg E G, Ugi Ugi: (m(g))((vg)) ((ggig)))))
(pgos(AD) Poo,i(h1) -
Cvee(W))= =
voc(W)) = 0. (20)
(14)
where Pyt and Pm are group representations of Ugs and Uno respectively.
(ho,((hM)- I
4.3. Equivariant Linear Weights
Note that the equivariant linear weight blocks (W)) Wib0=0 in the same subspace, which corresponds to the nullspace of the coefficient matrix C. We can obtain it via SVD.
J lie
The output feature Egoo E go is obtained by linear combination of the post-activation value Vmm E Un- Let U, = Rd: and Umo = Rdg=, then Equation (13) indi- cates that Um = I(G+++)dd; The linear weight ma- trix W E X(G+++I)) can be partitioned as W = (Wo W1 ... Wo+a]. where Wo E Xd: Combining with Equation (11), we have:
We summarize the architecture of the EKAN layer in Fig- ure (Right). In this example, the EKAN layer is equivari- ant with respect to a 2-dimensional matrix group G (such as the SO(2) group). The user specifies the input space U, To <b Ti (where we abbreviate T(p, 0) = V as Tp), which represents scalar space and vector space, and spec- ifies the output space U,, = T2, which represents matrix space. Then, the gated input space Ugi = To e Ti + To and the gated output space Ug = T2 Tu each add gate scalar the vector space Ti and the matrix space Tz. The basis functions are applied the gate scalars of To (itself) and Ti, which are then multiplied by the original terms to obtain the post-activation space Um = 3U, The linear weights W E RR5x9 between Um and Ugo are within the subspace determined by Equation (20) to ensure equivari- ance. Similar to KANs (Liu et al., 2024b), EKAN updates grids based on the input activations, which we discuss in detail in Appendix E.
G+*
Ugo = Um
WyUmb
(15)
b=0
To ensure the equivariance between the post-activation space and the gated output space, we obtain:
Vg E G Um E Um: ( = W pm (g) (16)
where Pgo is the group representation of Ugo Using the
structure of Um shown in Equation (13) and applying the rules from Equation (4), we can derive that (m(g) = 0 G+k P:(g), where Pi is the group representation of U..
Therefore, from Equation (11), we have Pm ( =
==0 (m(g) Then Equation (16) can be written
G+*
Vg E G. fomm
G+* E Us:
5. EKAN Architecture
G4*
G+4
In this section, we construct the entire EKAN architecture. Wopilg)um.b (17) The main body of EKAN is composed of stacked EKAN layers. The output space of the I-th layer serves as the input
Σ 6=0
Pgo Wo
Σ
6=0
5

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Incorporating Arbitrary Matrix Group Equivariance into KANS
final output label (after dropping)
Tg
gate (dropped)
output space Uo
gated output space Uo
x (N- 1)
latent space Up-
gated latent space Ug! x(N 1)
EKAN layer
XN
T
gate
user specified
gated input space Ugi
lift layer
dataset determined
Input space U,
data flow
To
T
control flow
raw input feature
Figure 3. (Left) The space structures of EKAN and their relationships. (Right) The EKAN architecture, which consists of lift layer and stacked EKAN layers.
space of the (I 1) layer, which we refer to as the latent space Ut. For the dataset, we usually know its data type, or in other words, how group elements act on it. To embed this prior knowledge into EKAN, we set the input space of the first layer the feature space of the dataset U, and the output space of the final layer as the label space of the dataset U..
equivariant architectures like EMLP (Finzi et al., 2021) and CGENN (Ruhe et al., 2023), EKAN achieves lower test loss and higher test accuracy with smaller datasets or fewer parameters. Additionally, for all trained models in this section, evaluate their equivariant errors in Appendix F,
6.1. Particle Scattering
However, the actual input/output features of the EKAN layer stack lie in UgaUUgo Therefore, we need to add extensions to align the network with the dataset. First, the gate scalars of the actual output feature are directly dropped to obtain the final output label of EKAN, which resides in U.. Then, we add a lift layer before the first layer to preprocess the raw input feature of EKAN, which is essentially an equivariant linear layer between U, and Ugs (see Section 4.3).
In electron scattering, can observe the four- momenta of the incoming electron, incoming muon, outgoing electron, and outgoing muon, denoted as qe, p", q" 5" E R4, respectively. We aim to predict the matrix element, which is proportional to the cross- section (Finzi et al., 2021): |M]2 x [p"r" - (p p"
- (g - qa))))uul According to Ein- stein's summation convention, in a monomial, if in- dex appears once superscript and once subscript, it indicates summation over that index. The metric ten- sor is given by 9uv = 9t = diag(1, -1, 11, -)), and an = an = (a", -a",-aa--- -aa. is invariant under Lorentz transformations. In other words, this task exhibits O(1,3) 3) invariance (see Appendix A.2 for more details), with the feature space U, 47, and the label space U, To. The data generation process for particle scattering is entirely consistent with that in EMLP (Finzi et al., 2021).
We summarize the space structures and network architecture of EKAN in Figure 3. The space structures of EKAN can be analogous to the dimensions of a conventional network. The user specifies the latent space U₁ of EKAN, which corre- sponds to specifying the hidden dimension in conventional network. The input space U, and the output space U, are de- termined by the dataset, similar to how the input and output dimensions are defined in a conventional network. In the concrete example, the feature space and label space of the dataset are U, To + Tı and U. T2, respectively. After passing through the lift layer, a new gate scalar is added to the raw input feature for Ti. resulting in the actual input feature Uns = To @ Ti @ To for the first EKAN layer. The gate scalar in the actual output space Ugo = T2 @ To of the last EKAN layer is dropped to obtain the final output label.
The matrix element
We embed the group ((1,3) and its subgroups SO* (1,3) and SO(1,3) equivariance into EKAN. Models are evalu- ated on synthetic datasets with different training set sizes, which are generated by sampling qt, p", q" f0 NN(0, }.). Both EKAN and KAN have the depth of L = 2, the spline order of & = 3, and grid intervals of G = 3. Although the lift layer increases the parameter overhead, we set the width of the middle layer in EKAN to n1 = 1000 (shape as [16, 1000, 1]. and the software will automatically calcu- late the appropriate feature space structure based on the user-specified dimension), and set the width of the mid-
6. Experiments
In this section, we evaluate the performance of EKAN on regression and classification tasks with known symmetries. Compared with general models such as MI.Ps, KANs, and
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Incorporating Arbitrary Matrix Group Equivariance into KANS
Table 1. Test MSE of different models on the particle scattering dataset with different training set sizes. We present the results in the format of mean + std.
Training set size
Models
102
1046
10a
1043
104
MLP
0,003 x 10
[6.97 0.09)
10
(3.61 (..1100.30) 10-
0.01 x 10-2
(1.66 +: 0.07) x 10
MLP+ + augmentation
(1.90 0.22) x 10
[3.97 + 0.52) x 10"
(1.25 = 0.08) 10-
(1.08 + 0.08) x 10-2
(1.34 + 0.22) x 10
(L,
(1.27 x 10-2
[2.21 + 0.56) x 10-3
(3.30 0.86) 10-1
(2.24 0.55) x 10-
(1.99 +: 0.33) x 10--
EMLP-SO(1.3)
0.91) x 10
[2.58 0.25) 10-3
3.69:11.25) 10
(2.73 x 10
(2.12 x 10
EMLP-O(1,3)
(8.88 2,51) x 10
(1.95 0.18) x 10-3
(3.30= 10
(2.66 x 10
(2.61 0.28 x 10
KAN
(6.70 1.35) x 10
10
(3.46 10"
(1.21 + 0.07) x 10
(2.57 + 0.08) x 10
KAN + augmentation
(5.98 = x 10-1
(4.72 1.52) x 10-1
(9.07 x 10-2
(5.61 = 0.70) x 10-2
(1.58 +: 0.06) x 10-1
EKAN-SO* (Ours)
(6.86 + 6.28 x 10-3
[1.85 + 1.75) x 10-3
(2.01 + 1.93) 10-3
(1.93 + 1.11) x 10-5
(4.29 + 3.38) x 10-8
EKAN-SO(1, 3) (Ours)
(6.86 +: 6.27) x 10
(1.85 + 1.75) x 10
(2.06 1.88) x 10
(2.17 = 1.11 x 10
(3.85 + 2.77) x 10
EKAN-O(1, 3) (Ours)
(7.77 +: 5.85) x 10-1
(1.64 1.87) x 10-3
(2.85 = 3.09) x 10-5
(7.31 # 1.15) x 10-6
(3.81 + 2.83) x 10-6
Table 2. Test MSE of different models with different numbers of parameters on the threc- bodyy problem datasct. We present the results in the format of mean std.
Number of parameters
Models
104.5
104-70
10
10
109.5
MLP
0.19) x 10-3
(4.70 x 10-3
(4.60 0.12) x 10-3
(4.17 0.24) x 10
(4.24+ x 10-3
MLP + augmentation
(9.43 + 0.88) x 10-3
(9.65 0.54) x 10
(1.01 0.09) x 10-2
(9.89 - 0.65) x 10
(9.91 + 0 x 10-3
EMLP-SO(2)
(2.28 1.17) x 10-3
(6.87 x 10-3
(3.55 + 1.59) x 10-3
(2.01 = 1.09) x 10-3
(5.34 x 10-3
EMLP-0(2)
8.71) x 10--
(1.18 x 10
(1,42 + 1.86) x 10-2
(7.37 = 7.60) x 10
(1.37 (1.77 x 10-3
KAN
(4.32 + 3.08) x 10-1
(2:21 + 0.65) x 10-2
(1.18 + 0.18) x 10-2
(1:23 = 0.34) x 10-2
(9:15 + 1.76) x 10-3
KAN augmentation
(7.94 0.15) x 10
(7.15 0.24) 10
(6.91 + 0.34) x 10
(6.88 = 0.06) x 10
(6.76 + 0.12) x 10
EKAN-SO(2) (Ours)
(1.12
10-3
(7.06 + 0.65) 10-4
(6.09 + 0.27) x 10-4
(4.26 0.19) 10
(4.84 + 0.68) x 10-4
EKAN-0(2) (Ours)
(1.48 : 0.37)
x 11
(1.12 = 0.24) 10
(7.91 (7.91 x 10
(6.06 = 0.36) 10
x 10
dle layer in KAN to n1 = 3840 (shape as [16, 3840, 1)) to keep the parameter count similar. Both EMLP and MLP have the depth of L 4 and the middle layer width of ni :22 = Vig 384 (shape as [16, 384, 384, 384, 1)). In these settings, EKAN (435k) has fewer parameters than EMLP (450k) and KAN (461k). We provide more imple- mentation details in Appendix G.
an analytical solution. Specifically, we observe the motion states of three particles over the past four time steps, denoted as {41, Pa1. 9:2- Prz, 9:3, Pis p and predict their mo- tion states at time L, denoted as (q Qr2 P12, Qra, pta)- Here, 9i E I²² and Pij E IR² indicate the position and momentum coordinates of the j-th particle at time i, re- spectively. The dataset contains 30k training samples and 30k test samples. When the input motion states are simul- taneously rotated by a certain angle or reflected along a specific axis, the output motion states should undergo the same transformation. Therefore, this task has O(2) equivari- ance (see Appendix A.1 for more details), with the feature space U; = 4 x 67; = 247' and the label space U, 67., The dataset for the three-body problem comes from HNN (Greydamus et al., 2019), and we note that they predict the motion trajectories of three particles, differing from the setup in CGENN (Ruhe et al., 2023), which addresses an N N-body problem involving five particles.
We repeat experiments with three different random seeds and report the mean + std of the test MSE Table 1. The results of EMLP and MLP come from the original paper (Finzi et al., 2021) under the same settings. Although EMLP performs better than non-equivariant models, our EKAN with different group equivariance further surpasses it com- prehensively, especially showing an urders-of-magnitud advantage on large datasets (training set size 2 10²). More- over, our EKAN with just 10 training samples achieves approximately 10% of the test MSE of EMLP with 10 training samples.
We embed the group O(2) and its subgroup SO(2) equivari- ance into EKAN and E.M... Both EKAN and KAN have the depth of L = 2, the spline order of k = 3, and grid intervals of G = 3, while both EMLP and MLP have the depth of L 4. The number of parameters is controlled by adjusting the middle layer width N for comparison (the shape of EKAN and KAN is [48, N, 12), while the shape of
6.2. Three-Body Problem
The study of the three-body problem on a plane (Greydanus et al., 2019) focuses on the motion of three particles, with their center of mass at the origin, under the influence of grav- ity. Their trajectories are chaotic and cannot be described by
7

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Incorporating Arbitrary Matrix Group Equivariance into KANS
Table Test accuracy (%) of different models on the top quark tagging dataset (Reomp = 3) with different training set sizes. We present the results in the format of mean = std.
Training set size
Models
10a
102-5
10
103-5
10*
Parameters
MLP
52.96 + 0.21
54.31 + 0.48
57.47 + 0.32
62.72 + 0.60
69.30 + 1.03
83K
MLP+ + augmentation
52.72 + 0.35
60.16 = 0.41
61.29 = 1.66
58.71 = 1.37
59.45 = 1.90
EMLP-SO*(1,3) (1,3)
65.48 + 1.21
72.59 0.84
74.40 + 0.26
76.34 + 0.14
77.10 + 0.02
EMLP-SO(1,3)
61,86 + 5.92
73.09 = 0.92
74.37 = 0.17
76.46 = 0.12
77.12 + 0.04
133K
EMLP-O(1,3)
62.66 1 7.35
73.65 1 1.01
74.22 i 0.53
76.26 1 0.05
77.12 1 0.04
KAN
49.89 + 0.39
49.91 + 0.43
49.89 = 0.37
50.00 = 0.02
49.84 = 0.25
35K
KAN + augmentation
49.83 + 0.28
50.09 + 0.09
49.73 = 0.40
50.02 + 0.01
50.27 = 0.67
CGENN
62.63 + 2.24
68.74 + 0.77
70.29 + 1.29
75.10 + 0.47
77.05 + 0.03
85K
EKAN-SO (1,3) (Ours)
71.92 + 0.88
73.98 + 0.39
76.15 0.11
76.69 + 0.08
76.93 + 0.02
EKAN EKAN-SO(1,3) 3) (Ours)
70.49 + 2.85
73.96 = 0.37
76.15
76. + 0.08
76.93 = 0.02
34K
EKAN-O(1,3) (Ours)
71.68 + 1.21
73.95 + 0.36
76.69 + 0.07
76.93 + 0.03
EMLP and MLP is [48, N, N. 12). More implementa- tion details can be found in Appendix G.2.
sets of different sizes from the original dataset for evalua- tion. Both EKAN and KAN have the depth of L = 2, the spline order of k = 3, and grid intervals of G =33. We set the width of the middle layer in EKAN to n2 200 (shape as [12, 200, 1)) and the width of the middle layer in KAN to n1 384 (shape [12, 384, 1) to control the number of parameters. Both MLP, EMLP and CGENN have the depth of L = 4 and the middle layer width n = n2 = R2g = 200 (shape as [12, 200, 200, 200, 1). We apply the sigmoid fune- tion to the model's output and use BCE as the loss function for binary classification. More implementation details are provided in Appendix G.3.
The mean + std of the test MSE over three runs with differ- ent random seeds are reported in Table Our EKAN-SO(2) and EKAN-O(2) consistently outperform baseline models with the same number of parameters, often by orders of mag- nitude. Notably, our EKAN with 10 parameters achieves comparable or even lower test MSE than baseline models with 105.5 parameters, saving 90% of the parameter over- head.
6.3. Top Quark Tagging
We report the mean + std of the test accuracy over three runs with different random seeds, as well as the number of parameters of the models in Table 3. Since we have not observed all the jet constituents, the relationship between the labels and input features cannot be accurately expressed as an explicit function. In this case of non-symbolic formula representation, KAN cannot achieve higher accuracy with fewer parameters than MLP as expected. On the other hand, our EKAN achieves comparable results with EMLP and CGENN using fewer than 40% of the parameters, improving test accuracy by 0.23% ~ 6.44% on small datasets (training set size < 104), while decreasing by 0.19% on large datasets (training set size- 104).
The research on top quark tagging (Kasieczka et al., 2019) involves classifying hadronic tops from the QCD back- ground. In particle collision experiments, top quark decays or other events produce several jet constituents. We ob- serve the four-momenta p/ P2 P/ E I of the three jet constituents with the highest transverse momentum pr. and predict the event label (1 for top, 0 for QCD). The category of the event will not change when all jet constituents un- dergo the same Lorentz transformation. Consequently, this task possesses O(1, 3) invariance (see Appendix A.2 for more details), with the feature space U₁ = 37, and the label space U.- = To. The top quark tagging dataset is sourced from Kasieczka et al. (2019), and we assume that only the Teomp = 3 jet constituents with the highest transverse mo- mentum Pr are observed, which differs from the setup in CGENN (Ruhe et al., 2023) and LorentzNet (Gong et al., 2022), where all nccomp 200 jet constituents are available.
7. Conclusion
To our knowledge, this work is the first attempt to com- bine equivariance and KANs. We view the KAN layer as a combination of spline functions and linear weights, and accordingly define the (gated) input space, post-activation space, and (gated) output space of the EKAN layer. Gated basis functions ensure the equivariance between the gated
Similar to particle scattering, we embed the group 0(1,3) and its subgroups S0* (1,3) and (0 3) equivariance into EKAN and EMLP. Furthermore, we embed ((1 equivari- ance into CGENN (Ruhe et al., 2023). We sample training
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Incorporating Arbitrary Matrix Group Equivariance into KANS
input space and the post-activation space, while equivari- ant linear weights guarantee the equivariance between the post-activation space and the gated output space. The prior work has demonstrated that "EMLP >> MLP" on tasks with symmetries and "KAN > MI.P" on symbolic formula rep- resentation tasks. Our experimental results further indicate that on symbolic formula representation tasks with symme- tries, "EKAN > EMLP" and "EKAN KAN". Moreover, on non-symbolic formula representation tasks with symme- tries, although it may be that "KAN < MLP", we show that "EKAN > EMIP" We expect that EKAN can become general framework for applying KANs to more fields, such as computer vision and natural language processing, just as EMLP unifies classic works like CNNs and DeepSets.
Pluim, J. P.,. and Duits, R. Roto-translation covari- ant convolutional networks for medical image analysis. In Medical Image Computing and Computer Assisted Intervention-MICCA. 2018: 21st International Confer- ence, Granada, Spain, September 16-20, 2018, Proceed- ings, Part I, pp. 440-448. Springer, 2018.
Bodner, A. D., Tepsich, A. S., Spolski, J. N., and Pourteau, S. Convolutional Kolmogorov- networks. arXiv preprint arXiv:2406. 13155, 2024.
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8. Limitations
Kolmogorov-Amold networks meet graph learning, arXiv preprint arXiv:2406. 18380, 2024.
One limitation of our approach is that the use of gating mechanisms can reduce the expressive power of EKAN, as discussed in Appendix D of EMLP (Finzi et al., 2021). However, due to the inherently complex structure of KANs (which involve intricate B-spline formulations), introducing symmetries into KANs is challenging. Therefore, for the sake of simplicity and clarity, we opt to incorporate gated non-linearities in a hierarchical manner. We anticipate that future improvements could enhance the expressive power and flexibility of EKAN
Cheon, M. Demonstrating the efficacy of Kolmogorov- Amold networks in vision tasks. arXiv preprint 2rXi::2406.14916, 2024a.
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Another limitation is that when data is sufficient, the ad- vantages of EKAN diminish compared with baselines. We provide additional experimental results in Appendix H to illustrate this point. This implies that EKAN enhances gen- eralization on small data but does not unlock the upper limit of expressive power.
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Acknowledgements
Dangovski, R., Jing, L., Loh, C., Han, S., Srivastava, A., Cheung, B., Agrawal, P., and Soljacie, M. Equivariant self-supervised learning: Encouraging equivariance in representations. In International Conference Learning Representations, 2022.
Z. Lin was supported by National Key R&D Program of China (2022DD0160300) and the NSF China (No. 62276004).
Impact Statement
Dauphin, Y. N., Fan, A., Auli, M., and Grangier, D. Lan- guage modeling with gated convolutional networks. In International Conference Machine Learning, pp. 933- 941. PMLR, 2017.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
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Incorporating Arbitrary Matrix Group Equivariance into KANS
A. Common Matrix Groups and Their Generators
A.1. Groups (2) and ((2)
The group S0(2) represents rotation transformations two-dimensional space. Its group elements can be expressed as:
cos 0
sin cos 0
R(0)
(21)
sin 0
It corresponds to infinitesimal generator:
1 0
A =
(22)
Then obtain the group elements through the exponential map R())) exp(0A1).
The group ((2) represents orthogonal transformations in two-dimensional space, including rotations and reflections. Based on the group SO(2), it has an additional discrete generator:
1
h1
(23)
0 -1
A.2. Groups SO+ (1,3), SO(1, 3), and ((1,3)
The group SO (1,3) represents Lorentz transformations that preserve both orientation and the direction of time. It includes six infinitesimal generators:
1
0
0
0
0
0
0
0
0
0
0
0
0
A
.A2-
AA3
0
0 0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
(24)
0
0
0
0
0
0
0
A4
AAs
As
- 0.
where A1, A2, Ag correspond to Lorentz boosts, and A4. As, Ag correspond to spatial rotations.
The group $0(1,3) represents Lorentz transformations that preserve orientation. Based on the group S0+(1,3) has an additional discrete generator:
0
0
-1 0
0
h1 =
(25)
0
-1
0
0
-1
which corresponds to orientation reversal.
The group 0(1,3) represents all Lorentz transformations. generator:
Based on the group SO(1,3), has an additional discrete
0
0
1
0
h2
(26)
0
0
0
0
0
1
which corresponds to time reversal.
B. Concrete Examples of Space Structure
First, let's intuitively understand the dual (+), direct sum (@), and tensor product (8) operations. Consider two vector spaces
X 2 = 3 and vectors x = (21, E X, = (y), 12, U3) E Y. Then, all x ① y (x1, F2, 81. U2, y3) form the
12

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Incorporating Arbitrary Matrix Group Equivariance into KANS
space X + Y R$, all y (119, m2y) 411y V3 #23/1 F22/2, T22/s) form the space X Y Re and all the coefficients ve)(W) of the linear maps W form the space Y X" TR.
Then, if we define how the group transformation acts on X, Y, the form of its action on these composite spaces can naturally be derived. Equation (4) provides the derivation rules. This paper assumes the group to be a matrix group. If a group element E G acts on a vector x E X in the form of its corresponding linear transformation gr, then we call the base vector space of
We have defined the "addition" and "multiplication" between spaces. Thus, for the base vector space of a group G. any complex space structure can be organized into the form of "polynomial" with respect to V, which is the origin of Equation (3). Note that Equation (3) simultaneously defines T(p q) VP @ (V*).. We abbreviate 7 (p, 0) as Tp
In the vast majority of scenarios, the feature space of a dataset takes simple forms such as vector stacking cTi = cV or matrices 2₂ = V @ V, while complex spaces like Tpp, q) rarely appear. However, the latent spaces between equivariant layers can be highly intricate (e.g., they may have 384 dimensions), and their decomposition forms with respect to the base vector space V often involve Tpp q) (the decomposition automatically handled by the software based on dimensionality, as described in Section 6). The practical implication is that, according the rules of Equation (4), we define how group transformations operate the latent spaces.
It is not correct that any (continuous, real, fnite-dimensionee representation U of matrix group can be written as in Equation (3). It can be shown (in the case of a compact Lie group for example) that U is subrepresentation of the direct sum on the right hand side. However, it is worth noting that, for the vast majority of practical applications, considering input/output representations of this form should suffice.
C. Equivariant Multi-Layer Perceptrons (EMLP)
EMLP (Finzi et al., 2021) embeds matrix group equivariance into MLPs layerwise. Given the input space U/ and output space Uo the linear weight matrix W E U. @ U. should satisfy Equation (5), i.e., Vg E G. E U (o( a = W (p((g)u. So the coefficients of each term in U2 equal Vgg E G Po(g)W Wpi(g). Flattening the linear weight matrix W into vector, we have Vy E G [p)(g) @ ( veo(W) = veo Combined with Equation (4), the linear weight matrix W is invariant in the space U. Uf:
Vg E Ō: Po:(g)vec(WW) = vec(W),
(27)
where Pa.i = @ p is the group representation of U, @ U,,. Decomposing the group representation (g) into discrete and infinitesimal generators as shown in Equation (2), Equation (27) is equivalent to the following constraint:
dpo:(AA)
dpo. Po, -
Cvec(W))=
vec(W) = 0.
(28)
Po., (hm)
By performing singular value decomposition (SVD) on the coefficient matrix C, we can obtain its nullspace, which corresponds to the subspace where the equivariant linear weights reside.
D. Proof of Theorem 4.1
Theorem 4.1. Given a matrix group G. the gated imput space Ugi and the post-activation space Um can be expressed in the forms of Equations (9) and (13), respectively. The function f Ugi - Um is defined by Equations (10) to (12), that is, Um = f(gg)) Then, f is equivariant:
Vg G Ugs E UgA: pm(g))((g))) ((prl)))g))
(14)
where Pgi and Pm are group representations of Ugi and Umn respectively.
13

---

Incorporating Arbitrary Matrix Group Equivariance into KANS
Proof. Let Um,b [b(Ugs), then Equation (11) be written as:
G+
(V = fo(w))).
(29)
b=ū
Using the structure of Um shown in Equation (13) and applying the rules from Equation (4), we can derive the group representation of Um:
G++
Pm (m)
P:(O),
(30)
b=0
where Pi the group representation of U. Combining Equations (29) and (30), we have:
G+k
Pm pm(g)f(Ug)) = θ Pe())fo(tg)).
(31)
O
Note that the group transformation in the scalar space To is the identity transformation, then we can obtain the group representation of Ugi from Equation (9);
Pgilg) Te
-
(o)
TA::
(32)
where Pi.a the group representation of T(pi,a: Ii,a). Therefore, applying the group transformation to Ugg in Equation (10)
results in:
A:
Pgc()) =
-
8i.0
+
-
((a(g)via
+
(33)
Substitute Equation (33) into Equation (12):
o - Si,a By (8i,a))
+ D Pi. (g) Bu Bulk.a)
G k, b G+
fu(Pg(())gg)))
(34)
O s n=l st,asilu(s
(t,all
Ai
Pi.o Ur si.ussi s/
Similar to Equation (32), derive the group representation of U, from Equation (8):
A
p((g) Ic +
Pr
(35)
Note that the right-hand side Equation (34) is the result of applying pr(g) to fo(Ugx), which means:
( = p(())So(vgg)
(36)
Substitute Equation (36) into Equation (29):
G++ + b=0
((Pg)())
pe(g)f6(1g)))
(37)
Combining Equations (31) and (37), Equation (14) is proven.
E. Grid Update
Similar to KANs (Liu et al., 2024b), EKAN updates grids based on the input activations. At the same time, the linear weights should also be updated in order to keep the output features unchanged. Let the post-activation values of the grids before and after the update be denoted as Vn. V N where is the number of samples. We first project the linear weight blocks into the equivariant subspace as W),( = Pvec(W,) and compute the output activations G++ V,,,W, where P is the equivariant projector obtained from Equation (20). Then, the updated equivariant linear weights W should satisfy Vgu = V WTT and we have W = VJJ (VTT)* We finally restore the updated linear weight blocks vec(W]) Piveo(W.)
Vyo T Ei=o
14

---

Incorporating Arbitrary Matrix Group Equivariance into KANS
F. Equivariant Error Evaluation
For all trained models fo in Section 6, we use Legui = The experimental results are presented in Tables 4 to 6, which indicate that our EKAN and EMLP can structurally guarantee strict equivariance, whereas non-equivariant models cannot. For partiele scattering, the results of EMLP and MLP are sourced from the original paper (Finzi al., 2021), so we do not present their equivariant errors here.
to evaluate their equivariant errors.
Table 4. Equivariant error of different models on the particle scattering dataset with different training set sizes. We present the results in the format of mean std.
Training set size
Models
10a
102.6
10a
10
10
KAN
(3.14+0.08) x 10
1.02 + 0.69
(4.88 + 2.02) x 10 1
(6.90 + 3.25) x 10
(1.64 + 0.41) x 10
KAN + sugmentation
0.30) x 10-1
(8.81 + 8.41) x 10-!
(1.08 + 0.55) x 1001
(1.32 + 0.52) x 10-1
(2.36 x 10-
RKAN-SO (1.3) (Ours)
(6.13 1.76) x 10-14
(9.32 + 2.75) 10
(7.48 + 1.61) x 100-4
(8.33 + 0.69) x 10-14
(6.57 + 0.06) x 10-14
EKAN-SO 3) (Ours)
(1.13 1.13:0.77 x 10-13
(7.61 1.67) x 10
(6,86 : 0.11) x 10-14
(7.98 + 0. 45) x 10-14
(6.02 4 0.89) x 10-14
EK EKAN-O(1,3) 3) (Ours)
(5.67 + 0.87) x 10-11
(7.95 + 2.67) x 10-14
(5.88 + 0.81) x 10-14
(9.00 + 0.63) x 10-14
(8. (8.43 + 0.18) x 10-14
Table Equivariant error of different models with different numbers of parameters on the three-body problem dataset. We present the results in the format of mean = std.
Number of parameters
Models
10
104-76
10a
105.25
105.5
MLP
(1.44 0.13) x 10-3
(1.51 + 0.14)
(1.54 + 0.03)
10-3
(1.33 x 10
(1.33 (1.33 0.10) 10
MLP + augmentation
0.362) x 10
(3.47 0.24)
10
(3.43 1 0.24) 10
(3.35 x 10
(3.32 1: 0.26) x 10
EMLP-SO(2)
(2.80 2.188) x 10-18
(1.57 = 1.39)
x 10
(2.17 0.37) x 10-14 14
(2.99 x 10-14
(1.87 x 10 10-14
EMLP-O(2)
(7.87 x 10-18
(1.87 = 1.32)
x 10
(3.22 -: 4.53) x 10-11
(1.46 x 10-12
(7.23 x 10
KAN
(3.00 x 10-1
(1.21 = 0.26)
10-2
(5.90 0.78) 10-3
(.93:1455 x 10-3
(4.03 x 10-3
KAN augmentation
(2.78 x 10-3
(2.49 = 0.11)
10-3
(2.37 0.11) 10-3
(2.35 x 10-3
(2.29 + 0.08) x 10-3
EKAN-SO(2) (Ours)
(3.80 3.84) x 10 13
(3.02 3.05)
13
(9.66 + 5.85) 10-14 14
(3.33 x 10 14
(3.31 1.31) 10-14 (7.62 4.91) x 10 14
EKAN-O(2) (Ours)
(9.79 6.13) x 10 13
(1.46 1.30)
10 13
(2.99 + 2.77)
(1.75 +: 1.55) x 10
Table Equivariant error of different models on the top quark tagging dataset (Reomp = 3) with different training set sizes. We present the results in the format of mean = std.
Training set size
Models
102
104.
10a
10a-5
10a
MIPP
0.40) 10-1
(4.00 0.17) x 10
(3.73 + 0.05)
10
(3.24 0.111)
10-1
(1.87
10
MLP sugmentation
(4.73
10
(2.15+ x 10
(1.90 0.55)
10
(3.73 0.399
10
(3.40 10
EMLP-SO
x 10-1
(1.09 x 10-7
(3.88
10-9
(3.54+2247) x 10-9
(1.65 x 10-
EMLP-SO(1,
x 10-7
(1.36
x 10-7
(5.49
10-9
(2.60
10-9
(1.71 x 10-9
EMLP-O(1,3)
(2.41 x 10-7
(1.86 x 10-7
(..20 10
(1.14 1.30) x 10
(1.26 0.28) x 10-9
KAN
x 10-1
1.77) x 10-1
(1.34 x 10
(9.90 + 14.00) x 10-5
(1.27 + 1.79) x 10-1
KAN augmentation
(1.20 = 1.61) x 10-1
(2.10+2297)) x 10-3
-.10011511 x 10-1
(1.52 + 2.15) x 10
(1.44 1.78) x 10-1
EKAN-SO (1,3) (Ours)
(3.41 2.99)
10-7
(1.66
10
(1.62 10-9
(1.65 1.01)
10-9
(1.64 1.53) 10-
EKAN-SO(L,3) (Ours)
(3.10
10-7
(1.09
10-8
(1.11 0.41) 10-9
(1.13 0.70) 10-9
(1.14 1.10) 10-9
EKAN-O(L,3) (Ours)
(3.14 2.32) 10-7
(1.46 10-8
(1.50 0.46)
10-9
(1.50 0.92) 10-9
(1.54 1.50) 10-9
G. Implementation Details
G.I. Particle Scattering
In particle scattering, we generate training sets of different sizes, and the corresponding test sets have the same sizes as the training sets. We train EKAN using the Adan optimizer (Xie et al., 2024) with the learning rate of 3 x 10-3 and the batch size of 500. For datasets with the training set size < 1000, we set the number of epochs to 7000, while for datasets with the training set size 2 1000, we set the number of epochs to 15000, which is sufficient for the MSE loss to converge to the
15

---

Incorporating Arbitrary Matrix Group Equivariance into KANS
minimum. We perform this experiment on single-core NVIDIA GeForce RTX 3090 GPU with available memory of 24576
MiB.
G.2. Three-Body Problem
In the three-body problem, we control the number of parameters by adjusting the middle layer width N of the model. We list the correspondence between the model's shape and the number of parameters in Table 7. We train all models using the Adan optimizer (Xie et al., 2024) with the learning rate of x 10-$, the batch size of 500, and for 5000 epochs. The grids of EKAN and KAN are updated every 5 epochs and stop updating at the 50th epoch. We perform this experiment on single-core NVIDIA GeForce RTX 3090 GPU with available memory of 24576 MiB.
Table The correspondence between the model's 's shape and the number parameters.
Number of parameters 10
Models
10a.5
104.75
105.25
105.5
MI.P
[48, 111, 111, 111,
[48, 153, 153, 153, 12
[48, 209, 209, 209, 12
[48, 283, 283, 283, 12]
[48, 383, 383, 383, 12] [48, 281, 281, 281, 12
EMIPP
[48, 84, 84, 84,
[48, 110, 110, 110, 12]
[48, 147, 147, 147, 12
[48, 214, 214,214, 12]
KAN
[48, 76, 12]
[48, 134, 12]
[48, 238, 12
(48, 423, 12]
[48, 752, 12]
EKAN (Ours)
[48, 45, 12]
(48,88, 88, 12]
(48, 151, 12
[48, 262, 12]
(48, 457, 12)
G.3. Top Quark Tagging
In top quark tagging, we train all models using the Adan optimizer (Xie et al., 2024) with the learning rate of 3 x 10-3 and the batch size of 500. For datasets with the training set size I 1000, we set the number of epochs to 1000, while for datasets with the training set size > 1000, we set the number of epochs to 2000, which sufficient for the BCE loss to converge to the minimum. The grids of EKAN and KAN are updated every epochs and stop updating the 50th epoch. We perform this experiment single-core NVIDIA GeForce RTX 3090 GPU with available memory of 24576 MiB.
H. Additional Experiments
For top quark tagging, we increase the number of observed jet components for further comparison. We present the results using the SO* (1, -equiivrari network as a representative, while the results of SO(1, 3) and ((, -qquivvraa networks are very similar. We the number of jet components to Reoop 10 and Reomp = 20 respectively, and the experimental results are shown in Tables 8 and 9. Together with Table in Section 6, note that all models exhibit significant improvements in accuracy the observed information increases. When Reomp is larger, EKAN achieves comparable results to the baselines, but it does not show superior performance. This suggests that EKAN may only have advantages in scenarios with insufficient observation information (as shown in Table 3), benefiting from its stronger generalization capability.
Table Test accuracy (%) of different models on the top quark tagging dataset (Neomp 10) with different training set sizes. We present the results in the format of mean + std.
Training set size 103
Models
102
102.5
103.5
104
EMLP-SO+ (1,
78.95 + 2.24
81.52 + 0.48
81.18 + 0.42
82.50 + 0.30
85.03 0.04
CGENN
71.82 + 3.28
80.35 + 0.57
79.85 + 1.01
81.56 + 0.23
84.17 + 0.76
EKAN-SO* + (1,3) (Ours)
78.57 + 0.63
79.77 + 0.78
82.90 = 0.61
84.83 + 0.19
87.14 + 0.03
16

---

Incorporating Arbitrary Matrix Group Equivariance into KANS
Table Test accuracy (%) of different models the top quark tagging dataset (Ncomp 20) with different training sizes. We present the results in the format of mean + std.
Training set size
Models
10a
102.3
10s
103.5
104
EMLP-SO* (1,3)
83.71 0.49
83.57 + 0.69
83.14 = 0.35
$5.00
86.81 + 0.14
CGENN
76.24 + 1.28
82.34 + 0.80
81.85 + 0.40
84.67 + 0.87
86.76 0.50
EKAN-SO (1,3) (Ours)
80.36 + 1.99
81.25 = 0.70
82.89 + 1.18
86.21 + 0.21
89.30 + 0.12
17


Reviews:
[{"review_id": "7IR1xeP6nd", "reviewer": "Reviewer_oRYZ", "summary": "The work introduces an equivariant version of the KAN by incorporating two principal components: 1) introducing an additional scaler that controls the gating mechanism and 2) using equivariant MLP for different non-scaler features.\n\nThe proposed model, EKAN, is evaluated on particle scattering, top quark tagging, and three body problem datasets, and it outperformed EMLP in almost all the scenarios across different train dataset sizes.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful reading and valuable feedback! Below we will address each of your concerns point by point.\n\n**Claims And Evidence**\n\nOur claim is: We propose a method to incorporate symmetry into KANs. As mentioned in Section 1 (Lines 25-29, Column 2), KANs struggle to respect symmetry, which is one of the reasons for their underperformance in non-symbolic representation tasks. By introducing equivariant layers, we aim to address this limitation. Experimental results show that EKAN outperforms KANs on symmetry-related tasks while preserving KANs' advantages over MLPs, which supports our claim. Furthermore, even in scenarios where KANs are weaker than MLPs, EKAN still surpasses EMLP—an insight we expect to provide valuable contributions to this emerging topic of KANs.\n\n**Experimental Designs Or Analyses**\n\n(1) The model architecture of EMLP is exactly the same as in the original paper [1]. To control the number of parameters, we adjust its depth and width (i.e., shape), with relevant details provided in Section 6 of the main text (specifically the second paragraphs of Sections 6.1, 6.2, and 6.3). For fairness, EKAN and all baseline models follow identical training settings, which we elaborate on in Appendix E (Implementation Details). We will emphasize this in the paper to avoid confusion. Thank you for pointing it out!\n\n(2) Overfitting often occurs with small datasets, so we reduced the number of parameters in EMLP and trained it on particle scattering with a relatively small training set size. The experimental results are shown below, which indicate that it does not perform as well as models with larger parameter sizes. In fact, such equivariant architectures are less prone to overfitting compared to non-equivariant models because they strictly respect the symmetries in the data.\n\n|Models/Training set size|$10^2$|$10^{2.5}$|\n|-|-|-|\n|EMLP-$SO^+(1,3)$ (210k parameters)|$(5.98\\pm5.40)\\times10^{-2}$|$(3.65\\pm1.60)\\times10^{-3}$|\n|EMLP-$SO(1,3)$ (210k parameters)|$(6.13\\pm5.62)\\times10^{-2}$|$(3.76\\pm1.73)\\times10^{-3}$|\n|EMLP-$O(1,3)$ (210k parameters)|$(6.14\\pm5.71)\\times10^{-2}$|$(3.64\\pm1.64)\\times10^{-3}$|\n|EMLP-$SO^+(1,3)$ (450k parameters)|$(1.27\\pm0.35)\\times10^{-2}$|$(2.21\\pm0.56)\\times10^{-3}$|\n|EMLP-$SO(1,3)$ (450k parameters)|$(1.47\\pm0.91)\\times10^{-2}$|$(2.58\\pm0.25)\\times10^{-3}$|\n|EMLP-$O(1,3)$ (450k parameters)|$(8.88\\pm2.51)\\times10^{-3}$|$(1.95\\pm0.18)\\times10^{-3}$|\n|EKAN-$SO^+(1,3)$ (435k parameters)|$\\mathbf{(6.86\\pm6.28)\\times10^{-3}}$|$(1.85\\pm1.75)\\times10^{-3}$|\n|EKAN-$SO(1,3)$ (435k parameters)|$\\mathbf{(6.86\\pm6.27)\\times10^{-3}}$|$(1.85\\pm1.75)\\times10^{-3}$|\n|EKAN-$O(1,3)$ (435k parameters)|$(7.77\\pm5.85)\\times10^{-3}$|$\\mathbf{(1.64\\pm1.87)\\times10^{-3}}$|\n\n(3) Our dataset for the three-body problem originates from LieGAN [2] rather than CGENN [3] (note that LieGAN is a method for symmetry discovery rather than the design of equivariant networks, so it is not our comparison target), so there are differences in dataset generation details and experimental setups. We supplement a comparison between CGENN and EKAN on the three-body problem, with the results shown below, which demonstrate EKAN's higher accuracy compared to CGENN.\n\n|Models/Number of parameters|$10^{4.5}$|$10^{4.75}$|$10^5$|$10^{5.25}$|$10^{5.5}$|\n|-|-|-|-|-|-|\n|CGENN|$(2.11\\pm0.11)\\times10^{-3}$|$(1.93\\pm0.40)\\times10^{-3}$|$(1.64\\pm0.21)\\times10^{-3}$|$(1.56\\pm0.18)\\times10^{-3}$|$(1.38\\pm0.26)\\times10^{-3}$|\n|EKAN-SO(2)|$\\mathbf{(1.12\\pm0.13)\\times10^{-3}}$|$\\mathbf{(7.06\\pm0.65)\\times10^{-4}}$|$\\mathbf{(6.09\\pm0.27)\\times10^{-4}}$|$\\mathbf{(4.26\\pm0.19)\\times10^{-4}}$|$\\mathbf{(4.84\\pm0.68)\\times10^{-4}}$|\n|EKAN-O(2)|$(1.48\\pm0.37)\\times10^{-3}$|$(1.12\\pm0.24)\\times10^{-3}$|$(7.91\\pm0.52)\\times10^{-4}$|$(6.06\\pm0.36)\\times10^{-4}$|$(6.02\\pm0.88)\\times10^{-4}$|\n\n**Other Strengths And Weaknesses**\n\nThe Background section primarily helps readers unfamiliar with symmetry theory to understand the relevant knowledge and avoid confusion. An important purpose of Eqn (4) is to inform readers that we decompose the space $U$ in the form of Eqn (3), whose practical significance is to define how group transformations act on $U$. This is crucial when constructing the latent space of the EKAN Layer (especially at the code implementation level), because the form of the group representation on the latent space depends on the definition provided by Eqn (4). More details can be found in point (2) of Other Strengths And Weaknesses section in the Rebuttal to Reviewer nHH7. In the revised version, we will move relatively less important concepts to the supplementary material. Thank you for your suggestion!\n\n**References**\n\n[1] Finzi et al. \"A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups.\"\n\n[2] Yang et al. \"Generative adversarial symmetry discovery.\"\n\n[3] Ruhe et al. \"Clifford group equivariant neural networks.\"", "claims_and_evidence": "The work is theoretically sound. The proposed architecture is equivariant with respect to the desired matrix group.\n\nHowever, the claim regarding the superiority of the equivariant KAN compared to other models is ill-demonstrated. \n\nThis, in the end, depends on the following two probable interpretations of claims made.\n\nClaim v1. **The work proposed an alternative to EMLP using KAN**: In this case, the experiments (with some additional details) support the claim. However, the scope becomes narrow.\n\nClaim v2. **The work proposed a new Equivariant architecture:** In this case, more experiments are required. For example, E(3) GNN[1]  or [2] should be considered.\n\nThe work should state the claim more precisely. \n\n[1]. Geometric and Physical Quantities Improve E(3) Equivariant Message Passing\n\n[2]. Scalars are universal: Equivariant machine learning, structured like classical physics", "methods_and_evaluation_criteria": "The method and evaluation criteria are valid. Depending on the interpretation of the claim (if Claim v1), the choice of dataset is also reasonable.", "theoretical_claims": "The theoretical claim is correct, to my understanding.", "experimental_designs_or_analyses": "I have found the following issues and uninvestigated questions:\n1. I do not find any details on the implementation of baselines.\n\n2. Why the number of parameters of EMLP is very high? Does it follow the architecture proposed in the original paper? is the performance gap is due to overfitting? What would be the performance if both EKAN and EMLP had a similar number of parameters?\n\n3. Why do we not consider equivariant GNN for three body problems? And it does not follow the exact setup of [a,b], i.e., five-body problem?\n\n\n\n\n[a] Clifford Group Equivariant Neural Networks\n\n[b] Geometric and Physical Quantities Improve E(3) Equivariant Message Passing", "relation_to_broader_scientific_literature": "KAN is an emerging area in machine learning. This work introduces equivariance to the KAN framework, which broadens its applicability. However, from the perspective of equivariant neural networks, most of the techniques used in this work are known results.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The work shows strong results compared to the EMLP baseline (some questions still need to be addressed). \n\nApart from limited novelty and evaluations, the writing of the paper can be significantly improved. For example, most of the topics discussed and notations introduced in the \"Background\" section are not used in Section 4 or in the main paper. For example, I do not believe the list in Eqn 4 is necessary to go through the main text. These can be moved to supplementary.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "No additional questions.", "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": "XwMqv8Mgjp", "reviewer": "Reviewer_nHH7", "summary": "This paper introduces Equivariant Kolmogorov-Arnold Networks (EKANs), a framework to construct group equivariant architectures, with respect to arbitrary matrix groups, as an extension of the previously proposed Kolmogorov-Arnold networks, akin to the way Equivariant MLPs (EMLPs) extend conventional MLPs. Contrary to the linear blocks that are the heart of MLPs, KANs contain several components (spline basis functions, as well as silu activations) that are non-trivial to be converted to equivariant analogues. To address this, in a nutshell, the authors follow a gating mechanism strategy (similarly to EMLP) - input scalars are gate-transformed and then given as inputs to the basis functions and silu which are subsequently scalar multiplied to the input tensors. In that way input features are transformed to post-activations, in a manner that is proven to be equivariant. Finally, post-activations are linearly transformed to the output features, where the linear equivariant parameters are calculated using a method obtained from EMLP. Empirically, the method is shown to achieve improved results on various symmetry-related tasks,  even with fewer parameters and/or training samples, against previously proposed equivariant models, as well as standard KANs and MLPs.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful reading and valuable feedback! Below we will address each of your concerns point by point.\n\n**Other Strengths And Weaknesses**\n\nWeaknesses\n\n(1) Indeed, the use of gating mechanisms can reduce the expressive power of the network. However, due to the inherently complex structure of KANs (which involve intricate B-spline formulations), introducing symmetry into KANs is challenging. Therefore, for the sake of simplicity and clarity, we opted to incorporate gated non-linearities in a hierarchical manner. We anticipate that future improvements could enhance the expressive power and flexibility of EKAN. We will include a discussion of this limitation and potential future work in the paper.\n\n(2) First, let's intuitively understand the dual ($\\*$), direct sum ($\\oplus$), and tensor product ($\\otimes$) operations. Consider two vector spaces $X=R^2,Y=R^3$, and vectors $x=(x_1,x_2)\\in X,y=(y_1,y_2,y_3)\\in Y$. Then, all $x\\oplus y=(x_1,x_2,y_1,y_2,y_3)$ form the space $X\\oplus Y=R^5$, all $x\\otimes y=(x_1y,x_2y)=(x_1y_1,x_1y_2,x_1y_3,x_2y_1,x_2y_2,x_2y_3)$ form the space $X\\otimes Y=R^6$, and all the coefficients $vec(W)$ of the linear maps $Wx=y$ form the space $Y\\otimes X^*=R^6$.\n\nThen, if we define how the group transformation acts on $X,Y$, the form of its action on these composite spaces can naturally be derived. Eqn (4) provides the derivation rule. The paper assumes the group to be a matrix group. If a group element $g\\in G$ acts on a vector $x\\in X$ in the form of its corresponding linear transformation $gx$, then we call $X$ the base vector space of $G$ (as described in Lines 102-105, Column 2).\n\nWe have defined the \"addition\" and \"multiplication\" between spaces. Thus, for the base vector space $V$ of a group $G$, any complex spatial structure can be organized into the form of a \"polynomial\" with respect to $V$, which is the origin of Eqn (3). Note that Eqn (3) simultaneously defines $T(p,q)=V^p\\otimes (V^*)^q$. We abbreviate $T(p,0)$ as $T_p$.\n\nIn the vast majority of scenarios, the feature space of a dataset takes simple forms such as vector stacking $cT_1=cV$ or matrices $T_2=V\\otimes V$, while complex spaces like $T(p,q)$ rarely appear. However, the latent spaces between equivariant layers can be highly intricate (e.g., they may have 384 dimensions), and their decomposition forms with respect to the base vector space $V$ often involve $T(p,q)$ (the decomposition is automatically handled by the software based on dimensionality, as described in Lines 317-320, Column 2). The practical implication is that, according to the rules of Eqn (4), we define how group transformations operate on the latent space.\n\nWe will add these intuitive explanations to the appendix to avoid confusion. Thank you for your suggestion!\n\n**Other Comments Or Suggestions**\n\n(1) Thank you for pointing that out! We will note this in the paper to be more rigorous.\n\n(2) We aim to express that $p_{i,a}$ and $q_{i,a}$ are not simultaneously zero. Indeed, in the case where both are natural numbers, $p_{i,a}^2+q_{i,a}^2>0$ and $p_{i,a}+q_{i,a}>0$ are equivalent, with the latter being more concise. We will make the corresponding revision. Thank you!\n\n**Questions For Authors**\n\n(1) Refer to point (1) in the Other Strengths And Weaknesses section of the Rebuttal.\n\n(2) When the dimension of the latent space between equivariant layers is too small, its parameters may degenerate to all zeros. However, this issue rarely occurs when the latent space is more complex.\n\n(3) Overfitting often occurs with small datasets, so we reduced the number of parameters in EMLP and trained it on particle scattering with a relatively small training set size. The experimental results are shown below, which indicate that it does not perform as well as models with larger parameter sizes. In fact, such equivariant architectures are less prone to overfitting compared to non-equivariant models because they strictly respect the symmetries in the data.\n\n|Models/Training set size|$10^2$|$10^{2.5}$|\n|-|-|-|\n|EMLP-$SO^+(1,3)$ (210k parameters)|$(5.98\\pm5.40)\\times10^{-2}$|$(3.65\\pm1.60)\\times10^{-3}$|\n|EMLP-$SO(1,3)$ (210k parameters)|$(6.13\\pm5.62)\\times10^{-2}$|$(3.76\\pm1.73)\\times10^{-3}$|\n|EMLP-$O(1,3)$ (210k parameters)|$(6.14\\pm5.71)\\times10^{-2}$|$(3.64\\pm1.64)\\times10^{-3}$|\n|EMLP-$SO^+(1,3)$ (450k parameters)|$(1.27\\pm0.35)\\times10^{-2}$|$(2.21\\pm0.56)\\times10^{-3}$|\n|EMLP-$SO(1,3)$ (450k parameters)|$(1.47\\pm0.91)\\times10^{-2}$|$(2.58\\pm0.25)\\times10^{-3}$|\n|EMLP-$O(1,3)$ (450k parameters)|$(8.88\\pm2.51)\\times10^{-3}$|$(1.95\\pm0.18)\\times10^{-3}$|\n|EKAN-$SO^+(1,3)$ (435k parameters)|$\\mathbf{(6.86\\pm6.28)\\times10^{-3}}$|$(1.85\\pm1.75)\\times10^{-3}$|\n|EKAN-$SO(1,3)$ (435k parameters)|$\\mathbf{(6.86\\pm6.27)\\times10^{-3}}$|$(1.85\\pm1.75)\\times10^{-3}$|\n|EKAN-$O(1,3)$ (435k parameters)|$(7.77\\pm5.85)\\times10^{-3}$|$\\mathbf{(1.64\\pm1.87)\\times10^{-3}}$|", "claims_and_evidence": "The authors' central claim is that their architecture is an equivariant extension of KANs that improves upon baselines (e.g. MLP) with fewer parameters and fewer data. Equivariance is proven theoretically, while the rest of the claims are indeed convincingly supported by experiments.", "methods_and_evaluation_criteria": "Since KANs have shown promise in other tasks, it is natural to extend them to symmetric problems. The benchmarks and evaluation criteria are obtained by EMLP, a known baseline in the literature. Therefore they are reasonable for the studied problem.", "theoretical_claims": "I studied the proof of Theorem 4.1, namely the equivariance of the proposed method, and I found it correct.", "experimental_designs_or_analyses": "As discussed above, the experimental designs follow those of EMLP and their implementations and analyses seem sound.", "relation_to_broader_scientific_literature": "Constructing group equivariant neural architectures has enjoyed a fruitful line of research in recent years, and a wide range of applications. Notably, the first paragraph of the related work section of the paper includes some of the most prominent results in this direction. On \nOn the other hand, Kolmogorov-Arnold networks are a relatively new architecture, serving as a promising alternative to  Multi-Layer  Perceptrons.  The method proposed in this paper attempts to address the poor performance of KANs on certain tasks, owing to their difficulty in respecting the data type and symmetry,  as per the authors, by incorporating group equivariance in the KAN framework for the first time.", "essential_references_not_discussed": "Nothing to note.", "other_strengths_and_weaknesses": "**Strengths**\n\n- The paper is well-structured and contains sufficient material on the background and related methods.\n- The equivariant conversion of KANs is easy to implement and widely applicable.\n- The empirical results demonstrate that the proposed method holds promise in real-world scenarios.\n\n**Weaknesses**\n\n- Apart from the improved experimental results provided in this paper, it is not evident how EKANs address the (i) scaling issues and (ii) limited expressivity (see Appendix D in Finzi et al., 2021) of Equivariant Multi-Layer Perceptrons (EMLPs) with gated non-linearities. Regarding (ii) it is unclear why the authors decided to resort to gating mechanisms for their construction and what are the implications of this choice.\n- Additionally, in certain parts the method is hard to follow for the reader not versed in symmetries, in particular due to the fact that the notation is quite specialised and not quite intuitive. I believe that some concepts need to be simplified with some indicative examples. For example, Eq. 3 could be explained with a concrete example. Similarly, I am confused with the notation T(p, q), since in the experimental section only T(p,0) spaces are encountered. Can the authors give concrete examples here as well?", "other_comments_or_suggestions": "- To my knowledge,  it is not correct that any (continuous, real, finite-dimensional) representation $U$ of a matrix group can be written as in Equation (3). It can be shown  (in the case of a compact Lie group for example)  that  $U$  is a subrepresentation of the direct sum on the right hand side. However, it is worth noting that, for the vast majority of practical applications, considering input/output representations of this form should suffice.\n\n- In lines 189-191, why are $p_{i,a}, p_{o,a}, q_{i,a}, q_{o,a}$ squared?", "questions_for_authors": "- Besides the provided experimental results, are there  any advantages  of  EKANs  against  EMLPs  with  gated  non-linearities?  Could their scaling issues or limited expressivity, somehow be addressed?\n\n- Does the linearity of the lift layer, between $U_i$ and $U_{gi}$, not prove problematic in some cases? For example, the only equivariant linear map from $T_1$ to $T_0$ is the constant zero, say, in the case of the orthogonal group (see Appendix D in Finzi  et al., 2021). This would mean that the gate scalars would always be zero in the gated input space.\n\n- In the  experiments,  would  an  EMLP  with  fewer  parameters  achieve  potentially improved results? Having more parameters than the EKAN would probably mean that it is a more expressive model, prone to overfitting phenomena.", "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": "CuDJEHyQ6N", "reviewer": "Reviewer_TCN4", "summary": "This paper introduces Equivariant Kolmogorov-Arnold Networks, an extension of KANs that incorporates equivariance to arbitrary matrix groups, addressing a key limitation of KANs: their inability to respect symmetries in data. The authors achieve this by constructing gated spline basis functions and equivariant linear weights, ensuring the model remains equivariant throughout. A lift layer is introduced to preprocess inputs, aligning them with the dataset’s symmetry properties.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful reading and valuable feedback! Below we will address each of your concerns point by point.\n\n**Claims And Evidence**\n\nFor all trained models $f_\\theta$ in the paper, we use $L_{equi}=E_{x,g}\\\\|\\rho_o(g)f_\\theta(x)-f_\\theta(\\rho_i(g)x)\\\\|^2$ to evaluate their equivariant loss. The experimental results are presented as follows, which indicate that our EKAN and EMLP can structurally guarantee strict equivariance, whereas non-equivariant models cannot.\n\nParticle scattering (for this experiment, the results of EMLP and MLP are sourced from the original paper, so we do not present their equivariance errors here):\n\n|Models/Training set size|$10^2$|$10^{2.5}$|$10^3$|$10^{3.5}$|$10^4$|\n|-|-|-|-|-|-|\n|KAN|$(6.01\\pm2.45)\\times10^{-1}$|$1.11\\pm0.63$|$(4.08\\pm0.27)\\times10^{-1}$|$(3.21\\pm0.27)\\times10^{-1}$|$(1.36\\pm0.14)\\times10^{-1}$|\n|KAN+augmentation|$(2.49\\pm0.12)\\times10^{-1}$|$(9.77\\pm7.74)\\times10^{-1}$|$(1.03\\pm0.14)\\times10^{-1}$|$(8.78\\pm1.13)\\times10^{-2}$|$(2.16\\pm0.10)\\times10^{-1}$|\n|EKAN|$(9.85\\pm3.12)\\times10^{-14}$|$(7.87\\pm1.47)\\times10^{-14}$|$(7.90\\pm0.74)\\times10^{-14}$|$(7.69\\pm0.48)\\times10^{-14}$|$(6.95\\pm0.70)\\times10^{-14}$|\n\n3-body problem:\n\n|Models/Number of parameters|$10^{4.5}$|$10^{4.75}$|$10^5$|$10^{5.25}$|$10^{5.5}$|\n|-|-|-|-|-|-|\n|MLP|$(1.44\\pm0.13)\\times10^{-3}$|$(1.51\\pm0.14)\\times10^{-3}$|$(1.54\\pm0.03)\\times10^{-3}$|$(1.33\\pm0.17)\\times10^{-3}$|$(1.33\\pm0.10)\\times10^{-3}$|\n|MLP+aug|$(3.16\\pm0.32)\\times10^{-3}$|$(3.47\\pm0.24)\\times10^{-3}$|$(3.43\\pm0.24)\\times10^{-3}$|$(3.35\\pm0.16)\\times10^{-3}$|$(3.32\\pm0.26)\\times10^{-3}$|\n|EMLP|$(2.80\\pm2.18)\\times10^{-13}$|$(1.57\\pm1.39)\\times10^{-13}$|$(2.17\\pm0.37)\\times10^{-14}$|$(2.99\\pm2.51)\\times10^{-14}$|$(1.87\\pm0.61)\\times10^{-14}$|\n|KAN|$(3.00\\pm2.28)\\times10^{-1}$|$(1.21\\pm0.26)\\times10^{-2}$|$(5.90\\pm0.78)\\times10^{-3}$|$(5.93\\pm1.45)\\times10^{-3}$|$(4.03\\pm0.81)\\times10^{-3}$|\n|KAN+aug|$(2.78\\pm0.10)\\times10^{-3}$|$(2.49\\pm0.11)\\times10^{-3}$|$(2.37\\pm0.11)\\times10^{-3}$|$(2.35\\times0.11)\\times10^{-3}$|$(2.29\\pm0.08)\\times10^{-3}$|\n|EKAN|$(3.80\\pm3.84)\\times10^{-13}$|$(3.02\\pm3.05)\\times10^{-13}$|$(9.66\\pm5.85)\\times10^{-14}$|$(3.33\\pm1.47)\\times10^{-14}$|$(3.31\\pm1.31)\\times10^{-14}$|\n\nTop tagging:\n\n|Models/Training set size|$10^2$|$10^{2.5}$|$10^3$|$10^{3.5}$|$10^4$|\n|-|-|-|-|-|-|\n|MLP|$(4.94\\pm0.40)\\times10^{-1}$|$(4.00\\pm0.17)\\times10^{-1}$|$(3.73\\pm0.05)\\times10^{-1}$|$(3.24\\pm0.11)\\times10^{-1}$|$(1.87\\pm0.03)\\times10^{-1}$|\n|MLP+aug|$(4.73\\pm0.28)\\times10^{-1}$|$(2.15\\pm0.12)\\times10^{-1}$|$(1.90\\pm0.55)\\times10^{-1}$|$(3.73\\pm0.39)\\times10^{-1}$|$(3.40\\pm0.63)\\times10^{-1}$|\n|EMLP|$(2.92\\pm2.30)\\times10^{-6}$|$(1.09\\pm0.84)\\times10^{-7}$|$(3.88\\pm1.44)\\times10^{-9}$|$(3.54\\pm2.47)\\times10^{-9}$|$(1.64\\pm1.53)\\times10^{-9}$|\n|KAN|$(1.36\\pm1.78)\\times10^{-1}$|$(1.35\\pm1.77)\\times10^{-1}$|$(1.34\\pm1.77)\\times10^{-1}$|$(0.99\\pm1.40)\\times10^{-4}$|$(1.27\\pm1.79)\\times10^{-1}$|\n|KAN+aug|$(1.20\\pm1.61)\\times10^{-1}$|$(2.10\\pm2.97)\\times10^{-3}$|$(1.10\\pm1.51)\\times10^{-1}$|$(1.52\\pm2.15)\\times10^{-4}$|$(1.44\\pm1.78)\\times10^{-1}$|\n|EKAN|$(3.41\\pm2.99)\\times10^{-7}$|$(1.66\\pm0.85)\\times10^{-8}$|$(1.62\\pm0.51)\\times10^{-9}$|$(1.65\\pm1.01)\\times10^{-9}$|$(1.64\\pm1.53)\\times10^{-9}$|\n\n**Other Strengths And Weaknesses**\n\nAs mentioned in the first paragraph of Section 5 (Lines 269-272, Column 2), EKAN is composed of stacked EKAN layers. The EKAN layer constructed in Section 4 has input space $U_{gi}$ and output space $U_{go}$ as shown in Equation (9). However, the feature space $U_i$ and label space $U_o$ of real-world datasets may not conform to this structure (specifically, they lack gate scalars). Therefore, as discussed in the second paragraph of Section 5 (Lines 295-299, Column 1), alignment is required. To align $U_{go}$ with $U_o$, we simply discard the gate scalars in $U_{go}$. For aligning $U_{gi}$ with $U_i$, we prepend a lift layer before the first EKAN layer to introduce gate scalars into $U_i$.\n\nThe lift layer is essentially an equivariant linear layer between $U_i$ and $U_{gi}$ (as described in Lines 300-302, Column 1). Its specific construction is detailed in Section 4.3, ensuring that it inherently preserves equivariance. Additionally, in the Claims And Evidence section of the Rebuttal, we also experimentally validate that the entire EKAN architecture is strictly equivariant.\n\n**Other Comments Or Suggestions**\n\nFigure 1 provides a general comparison between the EKAN and KAN architectures, which we elaborate on in the third paragraph of Section 1 (Lines 45-54, Column 2, and Lines 55-66, Column 1). Figure 2 illustrates the structure of an individual EKAN layer, which we explain in detail in the last paragraph of Section 4.1 (Lines 214-219, Column 1, and Lines 180-182, Column 2). Figure 3 presents the overall architecture of EKAN, which we describe in the final paragraph of Section 5 (Lines 304-318, Column 1). We will make it clearer to avoid confusion—thank you for pointing this out!", "claims_and_evidence": "The motivation for the work is clear, and the theoretical claims are supported by the work. The extensive experiments also show an improvement over the nonequivaraint version. However, there is no validation as to whether the network is actually equivariant (either through an equivariant loss, or through the study of the representations), and the equivariance is only indirectly evaluated through performance.", "methods_and_evaluation_criteria": "The methods and evaluation criteria are sensible for the task. As mentioned above, the paper would be strengthened by showing EKANs are actually equivariant.", "theoretical_claims": "I read the main theorem and it seemed reasonable, but did not thoroughly examine the proof. Similarly, I followed the derivations but did not thoroughly examine.", "experimental_designs_or_analyses": "The analyses are sensible and most observations seem insightful and accurate. Reiterating again my skepticism about the degree to which EKANs are structurally equivariant.", "relation_to_broader_scientific_literature": "KANs have been very popular in the past year and equivariance has been steadily impactful over the last decade, so the work is of very broad interest.", "essential_references_not_discussed": "There are no references (from the equivariance side) that I think are missing.", "other_strengths_and_weaknesses": "The motivation is very clear and the work is interesting. Evaluation of the degree to which the model is actually equivariant is the main weakness. Also, the lifting layer could be discussed in greater depth: why is it necessary, why does it maintain equivariance, etc.", "other_comments_or_suggestions": "There are 3 figures that showcase the EKAN architecture, all are different, and actually none of them is explained in the text.", "questions_for_authors": "No further questions, everything was addressed in the previous sections.", "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": "ZabS7DlByA", "reviewer": "Reviewer_yJMe", "summary": "This paper introduces Equivariant Kolmogorov-Arnold Networks (EKANs), an extension of Kolmogorov-Arnold Networks (KANs) that incorporates matrix group equivariance. The authors follow a similar approach as Equivariant MLPs (EMLP) by Marc Finzi et al. (2021) to enforce equivariance constraints on KANs. Basically, the weight is enforced to be in the intertwiner of the given group representations through a set of linear constraints. In addition, a lift layer is introduced to preprocess the input features, aligning them with the equivariant structure, while a gating mechanism ensures equivariance in the non-linear activation stage. The authors evaluate EKANs on tasks involving known symmetries (e.g., particle scattering, three-body problem, and top quark tagging), showing that EKANs achieve lower error and require fewer parameters compared to baseline models such as KANs, EMLPs, and other equivariant architectures.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful reading and valuable feedback! Below we will address each of your concerns point by point.\n\n**Claims And Evidence**\n\n> However, the key methodological innovation—incorporating equivariance via equivariant linear layers—is a direct adaptation of the approach used in EMLP (Finzi et al., 2021). The primary difference is that this method is now applied to KANs rather than MLPs. The authors claim novelty by applying equivariance to spline-based architectures, but they do not introduce a fundamentally new method for enforcing equivariance. This aspect is not sufficiently acknowledged in the paper.\n\nDue to the complexity of B-spline function formulations, embedding strict equivariance into KANs is challenging. Hierarchically employing gating mechanisms is a relatively straightforward approach, although we acknowledge this does have expressive limitations. Our claim is that we aim to introduce equivariance into KANs to address their poor performance on symmetric tasks and non-symbolic representation tasks, while preserving their advantages over MLPs. Thus, as an improvement to KANs, we strive to retain their original hierarchical structure and B-spline basis function form. From another perspective, if major structural modifications were made to propose an entirely new architecture, how would it remain related to KANs? Additionally, EKAN is not merely a simple fusion of KANs and traditional equivariant mechanisms—the design of the EKAN layer's post-activation space structure is also challenging (see Sections 4.1 and 4.2 for details).\n\n**Experimental Designs Or Analyses**\n\n> However, the paper does not explore settings where equivariance may not be useful or could degrade performance. It is noted in the EMLP paper that equivariant linear layers and gated nonlinearities are not universal, and there exist simple equivariant functions for some groups that cannot be approximated by EMLP. Since EKAN directly adopts the EMLP technique, it is likely that these limitations still exist and should be discussed in the paper.\n\nIndeed, the use of gating mechanisms can reduce the expressive power of the network. However, due to the inherently complex structure of KANs (which involve intricate B-spline formulations), introducing symmetry into KANs is challenging. Therefore, for the sake of simplicity and clarity, we opted to incorporate gated non-linearities in a hierarchical manner. We anticipate that future improvements could enhance the expressive power and flexibility of EKAN. We will include a discussion of this limitation and potential future work in the paper. Thank you for your suggestion!\n\n> Also, for the top tagging experiment, the authors have not included the comparison with GNN-based methods, e.g. LorentzNet (Gong et al., 2022), which generally have higher accuracies.\n\nThank you for your addition! LorentzNet [1] achieves good results in top quark tagging, but it specifically focuses on this particular task. EKAN aims to provide a general framework for all symmetric scientific problems, so methods targeted at downstream tasks are not our primary benchmark for comparison. For example, CGENN [2] can incorporate more symmetries beyond the Lorentz group and outperforms LorentzNet when observations are complete (with 200 components). Therefore, in our paper, we chose to compare with CGENN. As we claimed, our EKAN outperforms CGENN in both accuracy and parameter efficiency when observations are incomplete (with only 3 components and a smaller dataset). Combining EKAN with task-specific models remains a direction for future research.\n\n**Supplementary Material**\n\n> There is no instruction on how to run the experiments, and I did not review the code in detail.\n\nThe instructions for running the experiments are provided in the README.md document within the supplementary materials zip file. We will also make the code publicly available upon acceptance.\n\n**Relation To Broader Scientific Literature**\n\n> Overall, the paper positions itself as an application of existing equivariant principles to a new network architecture rather than a significant methodological advance. While the application to KANs may be useful, the extent of novelty is limited.\n\nRefer to the \"Claims And Evidence\" section in the Rebuttal.\n\n**Reference**\n\n[1] Gong, Shiqi, et al. \"An efficient Lorentz equivariant graph neural network for jet tagging.\" Journal of High Energy Physics 2022.7 (2022): 1-22.\n\n[2] Ruhe, David, Johannes Brandstetter, and Patrick Forré. \"Clifford group equivariant neural networks.\" Advances in Neural Information Processing Systems 36 (2023): 62922-62990.", "claims_and_evidence": "The main claim of the paper is that EKANs improve upon both KANs and other equivariant architectures by leveraging equivariance constraints in a more efficient manner. The empirical results support the claim that EKANs outperform standard KANs and EMLPs in terms of test error reduction and parameter efficiency.\n\nHowever, the key methodological innovation—incorporating equivariance via equivariant linear layers—is a direct adaptation of the approach used in EMLP (Finzi et al., 2021). The primary difference is that this method is now applied to KANs rather than MLPs. The authors claim novelty by applying equivariance to spline-based architectures, but they do not introduce a fundamentally new method for enforcing equivariance. This aspect is not sufficiently acknowledged in the paper.", "methods_and_evaluation_criteria": "The proposed modifications to KANs follow a well-established framework for imposing equivariance. The evaluation primarily focuses on:\n* Comparing EKANs against standard KANs, MLPs, and EMLPs on symmetry-sensitive tasks.\n* Measuring test error across different dataset sizes and numbers of parameters.\n\nThe benchmarks are chosen from EMLP and other related works. The chosen benchmarks are reasonable, and the paper provides sufficient details on experimental settings.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experimental results appear sound, and the comparisons are generally fair. EKANs are compared to multiple baselines, including MLPs, EMLPs, standard KANs, and other relevant equivariant architectures. Also, the study varies the dataset size and the model capacity to analyze their influence on performance.\n\nHowever, the paper does not explore settings where equivariance may not be useful or could degrade performance. It is noted in the EMLP paper that equivariant linear layers and gated nonlinearities are not universal, and there exist simple equivariant functions for some groups that cannot be approximated by EMLP. Since EKAN directly adopts the EMLP technique, it is likely that these limitations still exist and should be discussed in the paper.\n\nAlso, for the top tagging experiment, the authors have not included the comparison with GNN-based methods, e.g. LorentzNet (Gong et al., 2022),  which generally have higher accuracies.\n\n### References\n* Gong, Shiqi, et al. \"An efficient Lorentz equivariant graph neural network for jet tagging.\" Journal of High Energy Physics 2022.7 (2022): 1-22.", "relation_to_broader_scientific_literature": "This work heavily builds upon prior studies in equivariant networks and KANs. The foundational ideas stem from:\n* EMLPs (Finzi et al., 2021), which already established a general approach for incorporating equivariance into MLPs. EKANs directly adopt this approach for KANs without substantial modification.\n* KANs (Liu et al., 2024), which introduced spline-based learnable activation functions. The paper correctly identifies the limitations of KANs in handling symmetry constraints but does not present a fundamentally new way of addressing them.\n\nOverall, the paper positions itself as an application of existing equivariant principles to a new network architecture rather than a significant methodological advance. While the application to KANs may be useful, the extent of novelty is limited.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A\n\n## Post-rebuttal\n\nOne of my previous concerns is the novelty of this paper, compared to existing works such as EMLP. After careful reconsideration, I think incorporating equivariance into KAN is a valid and substantial contribution by itself, though the paper uses similar approaches to EMLP. I have updated my score to reflect this.\n\nIt should still be noted that the additional experiment results show that the advantage of EKAN diminishes with more data. EKAN may only be a better choice when data is relatively scarce.", "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:

Meta-review: This paper introduces Equivariant Kolmogorov-Arnold Networks (EKANs), extending KANs to incorporate arbitrary matrix group equivariance. After careful consideration of the reviews and author rebuttals, I recommend accepting this paper, although I note it is a borderline case. While concerns were raised about the novelty relative to existing equivariant architectures, the authors convincingly demonstrated that incorporating equivariance into the complex structure of KANs presents unique challenges that they've effectively addressed. The experimental results showing EKAN's superior performance with fewer parameters and training samples are compelling. We recommend the authors address the limitations related to expressivity and provide more detailed experimental comparisons in the final version.
Decision: Accept (poster)