FS-KAN: Permutation Equivariant Kolmogorov-Arnold Networks via Function Sharing
Authors: Ran Elbaz
Guy Bar-Shalom
Yam Eitan
Fabrizio Frasca
Haggai Maron
Venue: ICLR 2026 Poster (2026)
Keywords: Kolmogorov-Arnold Networks
Geometric Deep Learning
Group invariant neural network
Weight sharing
Parameter sharing
Symmetries
Primary area: learning on graphs and other geometries & topologies

Abstract: Permutation equivariant neural networks employing parameter-sharing schemes have emerged as powerful models for leveraging a wide range of data symmetries, significantly enhancing the generalization and computational efficiency of the resulting models. Recently, Kolmogorov-Arnold Networks (KANs) have demonstrated promise through their improved interpretability and expressivity compared to traditional architectures based on MLPs. While equivariant KANs have been explored in recent literature for a few specific data types, a principled framework for applying them to data with permutation symmetries in a general context remains absent. This paper introduces Function Sharing KAN (FS-KAN), a principled approach to constructing equivariant and invariant KA layers for arbitrary permutation symmetry groups, unifying and significantly extending previous work in this domain. We derive the basic construction of these FS-KAN layers by generalizing parameter-sharing schemes to the Kolmogorov-Arnold setup and provide a theoretical analysis demonstrating that FS-KANs have the same expressive power as networks that use standard parameter-sharing layers, allowing us to transfer well-known and important expressivity results from parameter-sharing networks to FS-KANs. 
Empirical evaluations on multiple data types and symmetry groups show that FS-KANs exhibit superior data efficiency compared to standard parameter-sharing layers, by a wide margin in certain cases, while preserving the interpretability and adaptability of KANs, making them an excellent architecture choice in low-data regimes.

Published as a conference paper at ICLR 2026
FS-KAN: PERMUTATION KOLMOGOROV-ARNOLI NETWORKS VIA FUNCTION SHARING
EQUIVARIANT
Ran Elbaz', Guy Bar-Shalom', Yam Eitan', Fabrizio Frasca', Haggai Maron Technion Israel Institute of Technology
2 NVIDIA Research
ABSTRACT
Permutation equivariant neural networks employing parameter-sharing schemes have emerged as powerful models for leveraging wide range of data symmetries. significantly enhancing the generalization and computational efficiency of the re- sulting models. Recently, Kolmogorov-
Networks (KANs) have demon- strated promise through their improved interpretability and expressivity compared to traditional architectures based on MLPs. While equivariant KANs have been explored in recent literature for few specific data types, principled frame- work for applying them to data with permutation symmetries a general con- text remains absent. This paper introduces Function Sharing KAN (FS-KAN), a principled approach to constructing equivariant and invariant KA layers for arbi- trary permutation symmetry groups, unifying and significantly extending previous work in this domain. We derive the basic construction of these FS-KAN layers by generalizing parameter-sharing schemes to the Kolmogorov-Arnolh setup and provide a theoretical analysis demonstrating that FS-KAN, have the same expres- sive power as networks that use standard parameter-sharing layers, allowing us to transfer well-known and important expressivity results from parameter-sharing networks to KANs. Empirical evaluations on multiple data types and sym- metry groups show that FS-KAN exhibit superior data efficiency compared to standard parameter-sharing layers, by a wide margin in certain cases, while pre- serving the interpretability and adaptability of KANs, making them excellent architecture choice in low-data regimes.
INTRODUCTION
Designing neural network architectures that respect symmetries has become a central theme mod- machine learning. with numerous works showing that leveraging symmetries lead im- proved generalization and computational efficiency (Cohen & Welling, 2016 Ravanbakhsh et al, 2017 Zaheer al. 2017 Esteves al., 2018 Kondor & Trivedi, 2018 Bronstein al., 2021 Maron et al. (019b). Equivariant models are designed so that certain transformations applied to the predictable transformation of the output. Permutation symmetries, subgroups of the symmetric group S, that act on a vector by permuting coordinates, are an important and natural family of symmetries that appear in real-world data. Although several methods exist, the most common, general, and scalable approach for designing equivariant architectures for permuta- tion symmetries is to encode equivariance each layer through a group-specific parameter-shie shraiig scheme (Wood & Shawe- Taylor 1996). These ideas have been applied across wide range of do- mains like images (LeCun et al. 1998). graphs (Maron et al. 2019b: Pan & Kondor 2022), sets
input correspond
Zaheer et al. 2017). and many more.
Networks (KANs) have recently been proposed as an alternative to tradi- tional Multilayer Perceptrons (MLPs), replacing scalar weights with learnable univariate functions [Lu et al.] 2024). This design is inspired by the Kolmogorov-Armol representation theorem Tikhomirov 1991) and leads to models with potentially improved expressivity relative to MLPs, particularly when the number of parameters is constrained. Some works have recently integrated
Kolmogorov-
KANs into equivariant models for graphs (Bresson et al. 2024 Kiamari et al. 2024), sets (Kashefi

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2024). and images (Bodner et al., 2024 In particular, variants like Graph-KAN (Kiamari et al., 2024 have achieved competitive or superior performance on graph-structured data. Very recently, Hu et al. (2025) proposed an approach for equivariant KANs, focusing on continuous groups.
While these approaches illustrate the potential of KANs for learning data with symmetries, the gen- eral principles for constructing equivariant KA layers for the important case of arbitrary permutation symmnetry groups remain unexplored. In particular, equivariant KANs have not yet been developed for numerous important data types that exhibit permutation symmetries, including multi-set inter- actions (Hartford et al. 2018), sets with symmetric elements (Maron et al., 2020). weight spaces
(Navon et [023] Zhou al., 2023), hierarchical structures (Wang et relational data (Maron et al. (019b).
2020 and high order
Our approach. In this work, we introduce Function Sharing KAN (FS-KAN), a general framework for constructing KANs with equivariant and invariant KA layers with respect to arbitrary permu- tation symmetry groups. Our construction generalizes the well-known parameter-sharing schemes used by previous equivariant architectures (Wood & Shawe-Taylor 1996] Ravanbakhsh al.) 2017 Maron et al. 2019bb by constraining KA layers to share functions rather than weights or parameters. In particular, arr, FS-KANs generalize several previously proposed equivariant KANs for sets (Kashefi, 2024) and images (Bodner et al., 2024). We discuss several typical function sharing scenarios that arise in natural data, like direct product symmetries and symmetries involving high-order tensors, and propose a more efficient FS-KAN variant to reduce computational and memory costs.
From theoretical perspective, we prove that FS-KAN architectures have equivalent expressive power to standard parameter-sharing MLPs in the uniform function approximation sense, implying that both architecture classes can represent the same functions. This equivalence establishes both fundamental limitations and guarantees for FS-KAN expressivity derived from equivalent results for crrammterr-hhainng neural networks, including expressivity analysis for parameter- shaeer--hhhin-bbaaa GNNs (Chen et al. 2019; Geerts 2020, Maron et al. 2019b a Azizian & Lelarge (0220) and univer- sality theorems for invariant or equivariant networks (ahherr et al. [017;] Yarotsky [2022] Segol &
Lipman 2019; Maron et al. 2019c Keriven & Peyré 2019) all of which directly transfer using
our result to their FA- counterparts.
Our experiments across multiple data types with varying types of permutation symmetries demon- strate that FS-KANs excel when learning on data with symmetries, achieving excellent parameter efficiency and significantly outperforming standard parameter-sharinn networks in low-data regimes. Moreover, we show that FS-KANs inherit the enhanced interpretability and adaptability of KANs, providing both transparent feature learning and robust performance in continual learning scenarios.
Contributions. (i) We propose FS-KAN, a principled framework for designing equivariant and invariant KA layers under arbitrary permutation symmetry groups using function sharing. (ii) We provide a theoretical analysis showing that FS-KANs match the expressivity of parameter-sharing MLPs and use this result to transfer well-known expressivity results to FS-KANs. (iii) We demon- strate empirically, across multiple domains, that FS-KANs excel when training data is limited, out- performing parameter-sharing MIPs and other natural baselines in data-scarce settings.
2 RELATED WORK
Equivariance in deep learning. A function is equivariant to a group action if a group transforma- tion of the input leads to corresponding transformation of the output. Perhaps the most well-known example of this principle is convolutional neural networks (CNNs) (LeCun et al. 1998) that employ translation equivariant layers; more recent works have generalized equivariance
broader symme-
try groups (Cohen & Welling 2016 Ravanbakhsh et al. 2017 Qi et al. 2017 Zaheer et al. 2017] Esteves et al. [0118] Kondor et al. [018]] Weiler & Cesal 2019 Pan & Kondor 2022] Hands al.,
2024 Zhang et al. 2024). For instance, Zaheer et al. (2017) characterized permutation equivariant layers in the context of set inputs, paving the way for many set-based architectures such as Hartford et al. (2018); Maron et al. (2020). Kondor et al.) (2018); MMaon et al.] (2019b a); |aan & Kondor (2022) studied equivariant layers for graphs, and Cohen & Welling (2016) introduced a framework for building networks equivariant to additional transtormations like rotations and reflections. Recent work by Navon et al. (202); Zhou et al. (2023) leverages symmetries within neural networks to de- velop effective model editing and analysis techniques. As constraining models to be equivariant might lead to models with limited expressive power (Xu et al., 2019 Morris et al.] 2019]] /eerts 2020), numerous works in recent years studied and characterized the expressive power of equivari-
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ant models (Zaheer et al. 2017 Maron et al. 2019a Dym & Maron 2020
Ravanbakhsh 2020
Azizian & Lelarge 2020
Kolmogorov-Armold Networks (KANs) and equivariant KANs. KANs Liu al. 2024 are recently proposed class neural architectures motivated by the Kolmogorov-
representation theorem (Tikhomirov 1991). They provide a promising alternative to traditional MLPs Ps due to their improved interpretability. adaptability, and parameter efficiency (Alter et al. 2024] Saaadi et al. 2024 Somvanshi al. 2024) and have demonstrated impressive performance (Xu al.) 2024 Moradi et al. 2024) Yang et al. 2024). Recent efforts to combine the strengths of KANS with the benefits of symmetry-awazt learning have led to several equivariant adaptations of KANs. On graph-structured data, [Bresson et al. (2024) integrates KAN into message-passinng passing frameworks and achieves competitive performance with respect to MI.P-based GNNs. Other works have introduced variants such as Convolutional KANs (Bodner et al. 2024). Graph-KAN (Kiamari et al. 2024), PoIN (Kashefi) 2024). Very recently, Hu et al. (2025) proposed equivariant KANs for continuous groups. While theoretically applicable to permutation groups, their approach requires solving numerically for equivariant layers compared to our straightforward efficient sharing schemes and, unlike our approach, cannot handle variable- sized inputs (e.g.. sets and graphs). To summarize, while prior works have developed effective equivariant KANs for specific permutation groups, we provide framework that unifies many of these works and establishes their theoretical foundations.
INVARIANT AND EQUIVARIANT FUNCTION SHARING NETWORKS
In this section, we formulate invariant and equivariant KA layers for symmetry groups G < Sn (section B.2). We then introduce variant called efficient FS-KA layer, which reduce both time and memory complexity. We provide concrete examples of FS-KA layers and demonstrate how our framework handles specific, yet important types of permutation groups including direct- product symmetries and high-order tensor symmetries (section 3.3 denote nn....,m..
We use the notation [x]] to
3.1 PRELIMINARIES
Kolmogorov-Arnold Networks. A KA layer 4b RT + R has the form of 4(x), Epm Pg (xp)) q E [Noat]. where 6g. IR + R are learned univariate functions. While the original KAN work uses splines to parameterize these functions, alternative parameterizations have also been explored (Li) [0224) (SS et al. 2024). Our construction this section does not assume a specific parameterization. Similar Liu (2024), we will use a Nout X Tin matrix of functions to represent the layers:
(.)
L1 (.)
I1
((x)
(1)
Dnea min())
Fna
Where the 'x' operator denotes applying the KA layer to the vector.
Invariance, equivariance, and standard parameter-sharing Let G S Sn be a group with representations (V,p), (V', A function L V + V is G-equivariant if it commutes with the group action, i.e., for all E G and v E V, L(p(g)o) = p' L(v). When using lin- layers L(u)= We, this equivariance constraint imposes structure the weight matrix W ('(x) W p(g) = W. Vi E G Wood & Shawe-Taylor, 1996. Ravanbakhsh et al. 2017 Maron al. 2019b Finzi et al. 2021) Specifically, when V
Rs and p p/ are the permutation representations of G < Sn, the weights W E RRax must satisfy the parameter-sharing condition to be equivariant: W.,- = W (i()).(()) Yo G. For example, convolutions, which are represented as circulant matrices, follow a parameter-sharing scheme for the cyclic group C,,- In the case of an in- variant layer T. R. -> IR, where the group acts trivially on the output space, invariance induces the following sharing scheme on the weight vector E "R w w Wo(s). Vo G. These constraints also generalize to higher-order vector spaces by the Kronecker product (Maron et al.) 2019b). These fundamental constraints form the basis of parameter-sharing in equivariant networks, as it defines principled approach for constructing equivariant linear layers. Equivariant and invariant nenworks can then be constructed using composition of parameter-sharin lincarities. Going forward, we simplify the notation of the permutation group action by omitting p. formally (p(g)x); (g x); 1 9 E G.
linear layers and point-wist non-
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3.2 INVARIANT AND EQUIVARIANT FUNCTION SHARING KA LAYERS
First, we define the Function Sharing (FS) scheme in the equivariant setting. Intuitively, and mo- tivated by standard parameter-sharing schemes, the idea is that the functions in the layer tied together according to the group action.
Proposition 1. Let be a KA layer with Thin = Now = nt. We say that & is a G-equivariant Function Sharing (FS) KA layer if and only if
24 (p))))a(p)) Vp. E [n], Yo E.
Moreover, any such G-equivariant FS-KA layer is G-equivariant (proof in Appx. B.2
This structure is conceptually similar to standard parameter-sharing: as illustrated in figure [ in their matrix form, linear convolution layers and KA-convolution layers (Bodner et al., 2024) have the structure of a circulant matrix. However, unlike the linear parameter-sharing case, we note that there exist KA equivariant layers that do not follow FS structure. For example, consider the following S2 equivariant KA layers:
cos() ) + 2 sin (.) - 2]
(cos()) sin(.)
sin(.) cos(.)
(()=
$0
sin ) + cos
Both layers compute the same function; however, only & is an FS-KA layer. As we will show next, every G-equivariant KA layer can nonetheless be represented by an FS layer: (proof on Appx. [B3)
Proposition 2. Let d be G-equivariant KA layer. Then, there exists G-equivariant FS-K KA layer such that (xx) = (x)), Vx E IR..
Importantly, this result allows us to restrict attention to FS layers when designing equivariant KANs without losing generality, simplifying both theoretical analysis and practical implementation.
Invariant layers. We define G- G-invariant FS-KA lay- ers as follows: (proof on Appx. [..4
V
W5
19g
142
15
A
V
Proposition 3. Let 4 be KA layer with nin n. Now = 1. If, vp E [n], Yo E G, Op Qn(p): then T is an G-invariant layer, and call G-invariant
35%
M4
My
W/s
V
0g
14g
12/
W/2
V
FS KA layer.
92
193
144
145
Similarly, the G-invariant FS-KA layers can express any G-invariant KA layer: (proof on Appx. B.5) B.5)
(a) Parameter-sharing (b) Function sharing
Proposition 4. every G-invariant KA layer has an equivalent G-invariant FS-KA layer representation.
Figure 1: Parameter-ssaring and FS b for C equivariant layers (ID-convolutional lay- ers). Function sharing constrains the func- tions be shared across the matrix accord- ing to the group action.
Feature channels. So far, we have considered in- puts with a single feature per element. When learn ing on data with symmetries, it is often useful to con- sider multiple feature channels, for example, the color channels in images. For the general case of multiple input and output features, we extend the formulation of KA layers for n elements with dm, dou features by:
(xx), = Σ 2 (xp),
g q [m], Ruxd,
pE(m)
Where each $T.P: Rd. - Rdu is KA layer. This creates matrix x n KA sub-layers, similar to decomposing linear layers into sub-matrices. We use similar formulation for the invariant case with nour = 1. We show that, in this case, the functions are externally shared (figure 2a) across corresponding positions in the DTP sub-layers: (proof (Bpx.[B66).
Proposition 5. Let & be a G-equivariant (invariant) KA layer with dan, dout input and oulput fea- tures. Then there exists KA layer & s.t 4, & represent the same function and the KA sub-layers satisfy: &... (p()) ($p= (o(w)), Yo G. We call & a G-equivariant (invariant) FS-KA layer.
We note that this FS pattern follows the same sharing scheme equivariant linear layers with multiple channels. These components allow us to define Function Sharing KA Networks (G- FS-KANs) as finite compositions of G-equivariant and invariant FS-KA layers.
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(a) Sm
(b) S. x Cm
(c) Sn. x Sm
Sy-equivariant (with d feature channels). bis S3 x Cs-equivariant (d = 1) and C is S3 x St equivariant (d = 1). The functions in each sub-layer in b are shared across the generalized diagonals (internal sharing w.r.t C4), while the sub-layers themselves are shared according to equation 5 (external sharing w.r.t S.).
Figure 2: Equivariant FS-KA layers for different groups.
Efficient FS-KA layers. In practice, standard KA layers can have significant computational and memory demands they apply functions independently across all input and output pairs. However, in many cases, the sharing structure requires the same operations to be performed across multiple elements, followed by sum aggregation. In linear layers, to reduce time and memory complexity, it can be helpful to commute sum or mean pooling with matrix multiplications. Inspired by this, introduce the Efficient FS-KA layer, which first aggregates imputs according to the FS structure and then applies shared KA sub-layer. While this relaxation does not yield equivalent layers, it reduces the number of nonlinear function applications while preserving equivariance, which can be easily verified. For example, consider S,,equivariant FS-KA layer (see equation [ It applies KA transformation independently to each element before performing sum-pooling over the set. In contrast, the efficient variant computes: ((x), $ (x,) क x,)) In this variant, sum over all elements so the second term can be computed once and reused (broadcast) for other computations within the layer. While the efficient FS-KA layer has the same number parameters, it reduces memory usage, especially during training, by retaining smaller computational graphs. For arbitrary permutation groups G, efficient FS-KA layers are derived following the same princi- ple: commuting element aggregations (sum mean pooling) with application of shared functions according to the original sharing structure. general, the efficiency of the layer is determined by the group structure. We discuss the efficiency and provide more formal construction of efficient FS-KA layers for general permutation symmetry groups in Appx. A.3
3.3 EXAMPLES AND IMPORTANT CASES
We now provide concrete examples of FS-KA layers and networks and demonstrate how they extend to important symmetry types. We first show how FS- FS-KANs generalize previous equivariant KANs, and illustrate their enhanced interpretability through a synthetic learning example.
FS-KANs generalize previous equivariant KANs. For set of d-dimensional vectors x E Raad equivariant 5, -E-FF-KANN are composed of S, equuivirri FS-KA layers in the form of:
(xx), (x,)) + Σ (2(x)),
(5)
where 11, 22 are shared KA sub-layers. We note that these layers are similarly structured to the DeepSets layers from Zaheer et al.] (2017) and can be seen as a generalization of PointNet-KAN (Kashefi) 2024). We illustrate the layer structure on figure 2a Similarly to the networks composed of equivalent linear layers, our S, SA-FS-KAN can process sets with varying sizes, and the sum-pooling can be replaced with other invariant aggregations, such as mean or max, which preserve equivariance and might work better in practice. Another example is M--convolution-ban based networks (Bodner et al. 2024), which are equivalent to our C,,FFSKAA KANS (Fig. lb).
Learning invariant formulas. KANs known for their superior interpretability through direct visualization of learned functions. To illustrate this in our case, we train both standard KAN and S, S.-NSSKAN to leam S,, invariant functions synthetic data, following the methodology Liu
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et al. 22024. Figure 3 presents a visualization of the trained networks. Due to the function sharing scheme, FS KANN learns shared spline functions across symmetric edges, making the equivariant structure immediately apparent and simplifying the learned network. In contrast, standard KAN learns independent spline functions for each edge, resulting in a more complex and less interpretable network that fails to respect underlying data symmetries. This demonstrates that FS-KAN not only maintains the interpretability benefits of standard KANs but enhances interpretability by making the equivariant structure explicit. We provide more examples in Appx. C.1
We now discuss specific instantiations of FS-KA layers to two key symmetry types: direct-product symmetries (Maron et al.) [2000] WWang et al.
2020) and high-order tensor symmetries (Maron
+
et al. 2019b c, Kondor et al. 2018 Keriven & Peyré
2019).
DS
FS for direct-product symmetries. For many data types that can be represented as matrices, the symmetry group can be formulated as di- rect product of groups G x H where G acts on the rows and H acts the columns. For exam- ple, in 2D images, symmetry exists in both the width dimension (horizontal shifts) and the height dimension (horizontal and vertical shifts), Simi- larly Maron et al. (2020), G H equivariant FS-KA layers have external sharing of KA sub- layers (WI.t G) and internal sharing of functions within each KA sub-layer (w.r.t to H). (iguluutbillsssrrttes for G = S, H = CT how functions are shared internally within each sub-layer, forming a circulant structure and effectively acting as 1 K KA-convolutions (Bodner et al. (024), while the sub-layers themselves ves are shared externally within the overall layer. In figure we visualize the case when = Sn and H = Sm. as encountered in user-item interactions (Hartford et al.) 2018). The formal construction and theoretical analysis are provided in
(a) Standard KAN
(b) FS-KAN
Figure 3: Visualization of learned spline func- tions for learning f(x) = exp(-xę x3 zg)). FS-KAN shares spline functions across symmetric edges (color-coded by function), with equivariant layers showing quadratic-like splines and the invariant output layer exhibiting exponential behavior, making the equivariant structure explicit and enhancing interpretabilityy.
ES for high-order tensors. In many real-world applications, input data naturally appears as ten- sors of different orders x E R²", and the group action is the diagonal action, (g x)i....ii (5-1(it)....a1661 For example, in graph-structured data, node features are first-order tensors, while connectivity can be represented as a second-order tensor. The KA layer formulation for layers map- ping k-order tensors to k'-order tensor can be readily derived from equation 4 and is given by: o(x)g EpE(N] (9.P(xp), q [m]] where each 4 dd - Rd are standard KA sub- layers. In this case, the natural equivariant FS structure can be derived from how the group acts on tensor indices $9.P = (p))) for all E G, where aii) (i.)...... (ix)). This construc- tion enables many important applications. For example, inspired by Maron et al. (2019b), we can build expressive GNNs Maron et al.) (2019a) as well as networks for hypergraphs with applications including co-authorship networks (Feng et al. [019 Yadati et al. (2019) and 3D mesh representation (Haji et al. 2022). We provide more details and discussion in Appx. A.2
4 A THEORETICAL ANALYSIS OF THE EXPRESSIVE POWER OF FS-KANS
While equivariant linear layers common building blocks, some of the resulting invariant and equivariant architectures, have known limitations - for example, cannot represent all invariant func- tions in the graph domain (Maron et al. 2019c Chen et al. 2019 Geerts 2020). In contrast, architectures like DeepSets (Zaheer et al. 2017 or PointNet (Qi et al.) 2017) are universal for permutation-invarre
functions for sets. In this section, we investigate the expressivity implications of choosing between FS- AAs and equivariant networks composed of equivariant linear layers. In particular, we show that for specific permutation group G. both model families have the same ex- pressive power in the uniform approximation sense. We begin by demonstrating that FS-KANS with splines activations can precisely implement any caaumtter-shaingg MLP (within a bounded domain).
Proposition 6. Let Q be bounded input domain. then for any parameter-sharin MLP f with I linear layers and ReLU activations, there exists FS-KAN & with at most 21 layers with splines activations such that: phat:((x) (xx,, E S.

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Proof idea. The key idea builds on Wang et al. (002b)), which shows that MLP layers with ReLU- like activations can be implemented using two KA layers: one for affine transformation and one for the point-wise ReLU. We decompose the parameter-sharee tion 4 each realizable by a KA layer. We use these realizations as the KA sub-layers, which com- pose equivariant KAA layer, and then apply Prop. obtain an equivalent FS-KA FS-KA layer. Adding a point-wise PS-KA layer for the nonlinearity gives two FS-KA layers per MLP layer. Composing these yields an FS-KAN that reproduces the parameter-sharing MLP. The full proof is on Appx.
linear layer into sub-layers in equa-
On the other hand, we will show next that parameter-sharing MLPs are capable of approximating FS-KANs arbitrary precision (proof is on Appx. B.8)
Proposition 7. For any compact domain Ω > O, and FS-KAN 4 with I layers and continuous activations,, there exists a parameter-sharee MLP fc with at most 21 ReLU layers s.t. |lf < c.
Both proofs provide methods for building the corresponding networks, though not necessarily effi- cient in terms of depth parameters. The results extend to higher-order tensors and direct products of symmetry groups via flattened tensor representations.
Implications for the expressivity of FS-KANs. We show that for specific permutation group G, FS-KANS and parameter-sharing MLPs can approximate each other arbitrarily well, meaning both model families have equivalent expressive power in the uniform approximation sense. This equivalence has profound implications for both theoretical analysis and practical applications. It allows us to transfer known properties and guarantees between these model classes, creating a bridge between the established theory of parameter-sharing networks and KANs. For example, if DeepSets proven universal for permutation-invarit FS-KAN construction inherits this universality. As direct implication of the above-mentioned expressive power equivalence, we establish the following corollary,
functions, we can establish that the corresponding
Corollary 4.1. FS- KANs with splines activations inherit the approximation properties of parameter- sharing networks:
1. They are universal approximators for translation- -equvvariinneuuuaaiiiinaaaa functions (as established for CNNs by Yarotsky (2022)) and permulation-eqqivariicnt functions over sets (derived from
Segol & Lipman (019)).
2. Higher-order FS- KANs are universal approximators for G-invariant functionns for any G S
Sn (Maron et al. 2019c
3. FS-KANs for graphs involving k-order tensors have the expressive power of k-Invariant Graph Networks (Maron et al. [019a Geerts 2020 Azizian & Lelarge 2020) and hence the same discriminative power as the k-WL test Morris et al. 2019)
EXPERIMENTS
In this section, we evaluate the effectiveness of FS-KAN by comparing it to standard parameter- sharing MLPs across a range of tasks involving data with symmetries. Specifically, we aim to answer the following questions: (1) Do FS- FS-KANs inherit the benefits of KANs? (2) How do FS- KANs perform compared to parameter-sheee sharing MLPs and non-equivariant architectures? (3) Do FS-KANs offer advantages in low-data regimes? and (4) What is the trade-off between efficiency and expressivity when using the efficient FS-KAN variant (section
Setup. We evaluated FS-KANs on di- verse invariant and equivariant tasks: sig- nal classification with multiple measure- ments (S,, x Cr), point cloud classifica- tion (Sn), and semi-supervised rating pre- diction (S, x Sm). These tasks involve structured signals, 3D geometry, and user- item interactions. We compare FS-KANs and efficient FS-KANs against parameter- sharing MLP baselines, matching parame- ter counts when possible as done in recent KAN literature (Liu et al. 2024 Bodner et al. 2024 Alter al., 2024), and additionally include transformers and standard KANs for point cloud classification. While GCNNs (Cohen & Welling
Male Tye
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(a) Low data regime
(b) High data regime
Figure 4: Test accuracy for signal classification with multiple measurements with varying train size.
2016) could serve as baselines, they are computationally infeasible for these tasks. Performance
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IS-KAN
Efficient KS-KAN
Point- TranttTrranformer
DeepSets
standard KAN
1024
0.533
0.533
0.011
3.677
D312
0767
0/6
D012
0.57
0.511
01572
0.514
0190
0,30
0322
0223
0.390
Test Accuracy
41007
10.00
61000
00:01
4:2
41000
41111
412
0.540
0500
0268
0012
0661
0120
100
0.73
0.32
0.68
0515
0006
002
0.21
0.21
00000
00:01
40000
+0000
4.07
4:21
0.533
0.530
0.400
0.550
3.603
0.000
0254
0013
0164
0600
0.433
0.73
08:3
0.533
0.77
1.400
0545
0/00
0.544
0.07
0. 0:33
0:37
0.44
0.217
0.77
66
8.537
0.570
0.636
3.67
0.000
0:46
0.000
0,27
0051
0.500
3.611
0.051
0.670
6.607
0/277
0/004
0.624
0.000
0.541
0.41
0.239
0.000
200
400
600
800
1000
200
4.0
500
800
1000
200
400
500
800
-000
200
400
600
500
1000
200
400
600
600
1000
Triin set 6428
Train aut 5922
Train and 6ize
Train sot si00
Triin not size
Figure 5: Test accuracy for point clouds classification with different numbers of n points in each point cloud and varying train set size. The highest accuracy is shown in red and bold. Our FS-KAN consistently outperforms DeepSets in all configurations.
evaluated across varying training set sizes to examine behavior under different data regimes. All experiments use 5 random seeds with mean results and standard deviation error bars. Detailed task descriptions, datasets, and architectures are in Appx. C with hyper-parameter study in Appx. D
Signal classification with multiple measurements We follow the signal classification setup from Maron et al. (2020), , using a synthetic dataset where each sample consists of a set of 25 noisy measurements of periodic ID signal sampled at T' = 100 time steps. The clean signals belong to one of three classes-sine, sawtooth, or square wave-with varying amplitude, frequency, phase, and offset. The task is to classify the signal type given the noisy measurements. For this experiment, we use S,, x FF----KN and its efficient variant, composed of S, CT equivariant and invariant FS-KA layers and batch-norms, with total number of 3e4 parameters. We compare our models to the parameter-sharing based Deep Sets for Symmetric elements model (DSS) proposed by the original work, which contains about 3e6 parameters and scaled DSS with comparable number of parameters to FS-KAN, Figure Figure4shows 4shows the classification accuracy across different training set sizes. FS-KA KAN consistently outperforms both bascline models in the low-data regime (60-1200 1200 examples) while yielding comparable results in higher-data regimes. The efficient variant achieves even higher accuracy than the full FS-KAN model while also reducing training time and memory usage. The training of the efficient variant was x 1.4 faster than that of the full FS-KAN. However, the efficient variant was about times slower than the DSS models 3
Point cloud classification We evaluate our models on the task of 3D object classification using the ModelNet40 dataset (Wu et al., 2015). which contains point cloud representations of objects from 40 categories such as chair, table, and airplane. Each sample consists of a set of n points with d = 3 spatial coordinates. We evaluated the models' ability to generalize by varying the training set size and also examined how the number of points in each cloud affects their performance. We do not use data augmentations to ensure that the number of training examples reflects the actual dataset size. We compare -AAN and the efficient variant to parameter-sharing based MLPs, composed of DeepSets (Zaheer et al. 2017) equivariant and invariant layers, as well as Point Transformer (Zhao et al. 2021) and a non mnvniinvriiant KAN baseline. Each invariant model has approximately 5.5 x 10* parameters while the non- non-invariant KAN has varying parameter counts due to different imput sizes. As observed in the previous experiment and supporting our hypothesis, FS-KAN outperforms the other models when the data is limited in both the number of samples and the number of points for each object, as depicted in figure 5 Consistent with previous works, the non-invariant KAN exhibits dramatically worse performance, highlighting the importance of designing symmetry-aware architectures. Additionally, we measure the runtime of each model in both training and inference (Table as well as memory usage during both phases (Table 5 While the efficient variant is about 1.5 faster than the FS-KAN training with larger n, it still slower than DeepSets.
Continual learning on point clouds In real-world applications, data distributions frequently shift over time, necessitating models that can continuously adapt while preserving previously acquired knowledge (Wang et al. 2024a Kirkpatrick et al.) 2017). A critical challenge is catastrophic for- getting, where adapting to new distributions degrades performance on previous ones. To evaluate the robustness of our FS-KANs against catastrophic forgetting, we conducted continual learning experiment following the methodology established in the original KAN paper and Park et al. (2024). Our experimental design comprises two phases: Phase I trains on the original ModelNet40 dataset (Wu et al., 2015) for point cloud classification (task A), while Phase 2 trains on a corrupted version featuring random translations and 3DD rotations (task B). Further details appear in Appx. C.4 We
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Train Size
Model
sec. AT
ncc. A2T
aec. B2
Forgetting
Avg Accuracy 1
200
FS-KAN
0.531 = 0.007
0,497 = 0.014
0.343 = 0.014
0.034 0.018
0.420 = 0.013
200
Efficient FS-KAN
0,511 + 0.011
0.470 0.018
0.320 + 0.009
0.040 + 0.010
0.395 + 0.012
200
DcepSets
0.514 0.009
0.455 : 0.005
0.304 = 0.019
0.059 = 0.014
0.380 + 0.009
400
FS-KAN
0,594 s: 0.008
0.550 0,010
0.374 = 0.014
0,044 0,007
0,462 0,012
400
Efficiont FS-KAN
0.567 = 0.004
0.506 : 0.019
0.345 = 0.015
0.061 0.020
0.425 + 0.013
400
DeepSets
0,562 0.005
0.495 0.019
0.344 a 0.018
0.067 0.017
0.419 0.018
600
FS KAN
0.635 = 0.005
0.390 0.014
0.411 z 0.004
0.045 = 0.013
0.501 = 0.006
600
Ellicient PS-KAN
0.613 0.008
0.553 0.014
0,384 a 0.011
0.060 0.012
0,469 0.012
600
DcepSets
0.614 0.007
0.559 0.011
0.391 = 0.006
0.055 0.012
0.475 + 0.007
800
FS. KAN
0.661 0.006
0.623 + 0.013
0.446 = 0.012
0.038 + 0.010
0.535 0.011
800
Efficient FS KAN
0.643 a: 0.007
0.586 a: 0.020
0.407 =: 0.015
0.056 0.022
0.497 = 0.016
800
DeepSets
0,641 0.014
0.606 0.005
0.427 0,008
0,036 0.011
0.516 0.006
1000
FS-KAN
0.679 + 0.007
0.640 = 0.007
0.467 = 0.011
0.040 = 0.006
0.553 + 0.009
1000
Efficient FS-KAN
0.666 0.005
0.615 0.003
0.435 + 0,010
0.051 0.006
0.525 0.005
1000
DeepSets
0.670 &: 0.006
0.643 a 0.016
0.468 = 0.017
0.027 + 0.020
0.555 + 0.014
Table 1: Test accuracies and continual learning metrics on point cloud classification for different models and training set sizes, Best results are in bold.
evaluate performance using two metrics: Forgetting, calculated the difference between accuracy on task A after Phase 1 and after Phase 2 (acc. Al acc. A2), and Average Accuracy, computed as the mean performance across both tasks after Phase Lower forgetting indicates better retention of previously learned knowledge, while higher average accuracy reflects robust overall performance. The results in Table [ show that FS-KAN demonstrates strong performance across training sizes, with clear advantages in low-data regimes where it consistently outperforms baselines.
Semi-supervised rating prediction We evaluate FS-KAN a matrix completion task under ex- treme data scarcity. Given partially observed user-item rating matrix, the goal is to predict the missing ratings. This setting exhibits a symmetry under both user and item permutations, which we model using S. X Sm equivariant architectures. We run experiments on DoviieLnns-OOO,, (Harper & Konstan 2015) dataset and the sub-sampled Flixster, Douban and Yahoo Music presented by Monti al. (2017). To simulate sparse supervision, train varying small fractions of the avail- able training data. S. x Sm FA- and its efficient variant are compared to the Self-supervised Exchangeable Model (SSEM) proposed by Hartford al. (2018) (2e6 parameters), composed of S, x Smn equivariant parameter-sharing layers, and scaled SSEM with the same number of param- eters as in FS- KS- (9e4 parameters). The data preparation procedure is detailed in Appx. C.5] As shown in figure 6 FS-KAN constantly achieves higher accuracy than the baselines in the low-data regime, highlighting its strong data efficiency. On larger datasets, the performance gap narrows, and models become preferable due to their shorter training time.
the standard linear
Discussion Our experiments show that FS FS-KANs outperform parameter-sharing MLPs low-data regimes while using fewer parameters, consistent with recent
1.38
1.12
1.5
1.10
1.50
100
1.2
1.0
1.64
KAN findings (Liu et al., 2024, Kiamari et al. (0224). Additionally. show that our FS-KAN inherits the bene- fits of KAN, such interpretability (fig- ure 7 7 and adaptability, while maintain- ing the benefit of incorporating symme- try. The efficient FS-KAN variant offers improved speed but remains slower than MLP baselines. These results suggest FS- KANs are most valuable for learning tasks on data with symmetries when data is lim- ited, where their superior sample efficiency justifies the computational overhead.
1.00
100
1.00
1.00
experiments
1.34
1/22
300
433
503
66C
200
93)
200
301
400
603
300
P00
Naxtoy of vaiile mntiee in Donning
Narde of 1ull
(a) Yahoo-Music
(b) Flixster
1.20
1.15
PS-KAN
Madd Type
1.7
1.5
16
3.14
1.00
1.17
1.00
S5EM
1.12
035
1.11
(000
1.10
035
1300
100
206
703 300
Nurther of valll ralngs maiiiit
Nerbor (148000 oringe Enslliing
(c) Douban
(d) MovieLens- 100K
Figure 6: Test RMSE comparison across different recommendation datasets. Our FS-KAN models out- perform baseline approaches in the low data regime, demonstrating better data efficiency and generaliza- tion capability.
6 CONCLUSION
In this paper, we presented a novel ap- proach to permutation-equivariai learning through KAN-based architectures. Our the- oretical framework unifies several recent
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architectures in this domain while providing new deep insights into their expressivity and offering blueprint for designing invariant and equivariant KAN- based networks for numerous other symme- try groups eurrently lacking in the literature. While performance on tasks with limited data, it has several limitations. For example, the computational cost be significant data regime. Hence, an important avenue for future work is to come up with faster implementations FS-KASS Finally, while our theoretical results focused mainly on expressivity, exploring proper- ties of FS FSSK,,,, such as generalization power, optimization issues, and scalability, is an important future research area.
proposed method demonstrates improved
with the efficient variant. This becomes especially apparent in the high-
ACKNOWLEDGMENTS
HM is supported by the Israel Science Foundation through a personal grant (ISF 264/23) and equipment grant (ISF 532/23), and by the Career Advancement Chairs in Artificial Intelligence Schmidt Futures. G.B. is supported by the Jacobs Qualcomm PhD Fellowship. FF conducted this work supported by an Aly Kaufman Post-Doctoral Fellowship.
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A GENERALIZATIONS OF FS-KA LAYERS
A. GENERALIZATION DIRECT PRODUCT SYMMETRIES
There are many cases where the symmetry of the data can be described using the direct product of groups x H. For example, in the case rating matrix X E gm permutation of the users and a permutation of the items will lead to corresponding permutations of rows and columns of the rating matrix. In general, for < Sn < S. the group G x acts on X E R x via:
((, T) X)1s X, 10)---00 '0)
(0,T) e G H
(6)
The standard KA layer in this case can be formulated as:
2xx), = Σ 2.. (..P(xp))
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Where each 47 is KA sub-layer from Rmxda to Ru As stated, G x H FS-KA layers can be constructed to be equivariant using FS-KA sub-layers as follows:
Proposition 8. Let 4 Ruxxmxd + Ruxmxdu be H equivariant FS-KA layer. Then exists a KA layer & composed of H-equivariant FS-KA sub-layers ($9.)) Jqwetna Jape(n) with din, dowr that satisfy:
haP )0(1)))0))
E
(8)
and,
(x)x) - =0(x)a- Vx
(9)
The proof is on Appx. B.9
A.2
HIGHER-ORDER TENSORS
As stated in section 3.3
the equivariant FS-KA layer for higher-order tensors satisfies:
$ (0(g))o(p)
Ya E G.
(10)
FS-KA layers this form can represent any other equivariant KA layer. The proof similar the first-order case, except it applies to k and k' e'-tuples of indices. When the data involves mixed-order tensors, we can naturally use a superposition of FS-KA layers:
+(x) = Σ R (xk),
E
(11)
where Xk the k-order input tensor, 2(x) is a k' -order output tensor, and & A an FS-KA layer mapping from to oo-order tensors.
A.3 EFFICIENT KS-KA LAYERS
In this section we will explain how derive the efficient equivariant FS-KA layer for arbitrary group G < S,. This derivation can be readily extended to invariant layers or to layers operating on higher-order tensors. general, we can write the equivariant FS-KA layer output as follows (using
Lemma B.2
(x)x = Σ Σ p=1 [NENNP/N
(la (x))1()))))))) E Or}
Σ Σ p= [
(x,)) Orb.
(12)
where [[n]+/G] is the orbits of [n]² under the group action of G and I is the indicator function, and (1)) KA sub-layer shared across orbit Oμ- Our Efficient FS-KA layer performs sum-pooling before applying KA sub-layer,i.e: ie::
((x),
Σ
(1b)
a1,9g + Σ II(((a))) E EOuD).
(13)
OnElIn)//G
p=1
where an E {0, 1} is for flexibility in the purpose of reusing computations (see the following exam- ples).
While this relaxation does not have the expressivity guarantees of the FS-KA layer, it is still equiv- ariant and can reduce the number computations and memory costs.
Equivariance. We will show that the efficient variant is still G-equivariant. Let G, then:
① x)
Σ
((h)
anX 1(0) Σ x H() E
On
(14)
Σ
(b)
OnX2 1(x) Σ x (() EO))
O.E[[n]//G]
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By the definition of the orbits, (g.p) E On iff (α(g), On Therefore,
Φ x) = Σ
((h)
on 1(1)
xpl ( (0).p) EO.))
O^E([n|2/a]
(15)
= (a (xxx(
Complexity analysis. If the same orbit indicator not zero for pairs P1), (g. P2), they must lie in the same orbit under the group action. Then, there must exist permutation in G that maps pi pa while leaving unchanged. In other words, the relevant orbits of [n]² are determined by the action of the stabilizer subgroup (t) the set of elements in G that fix q. Thus, for each q, the number of times ((() is applied in the efficient layer corresponds to the number of equivalence classes in [n|/Sun (g/ttaba(g).
As result, the total number of function evaluations required by the Efficient FS-KA layer most:
dindou
[ml/ (n]/Staba(g)) S n² din
(16)
However, this is only an upper bound, as some computations can be reused. We will show how efficient this variant is in some examples.
Examples. Consider G = Sn x Sm. In this case, we have four orbits, and the FS-KA layer can be expressed as
(())1 = = + A
(x + Eds(xis)
Σ
Ds(xL.))
(17)
For 02 a3 a4 1 the efficient FS-KA layer has the form of:
(x)):
1 (Xi,)) + &2
Σ
Xi,k
+ $
Σ 1=1
XLJ
th
X1,
(18)
1=
As the second and third term can be broadcast across the rows and columns respectively, and the fourth term is shared across all the layer, the layer performs only (n + 3)d complex computa- tions.
We would also examine the case of G = Sn x Cr. this case, we have 27 orbits, and the layer can be expressed
(xx);, Σ (1.T) (x) + Σ (2.T) (XK,tr)),
(19)
where ((1,r), ((2,7) dd + Rd are shared KA sub-layers. We can reduce the number of distinct computations in the layer by a factor of 2n by,
a+1
n
(x)).. Σ (( (x..
Σ ((2.)
Σ
(20)
In other words, the layer applies KA-convolution to each element and KA-convolution to the sum of all the elements in the set.
B PROOFS
B. KEY SUPPORTING LEMMAS
We state and prove two lemmas that will be used throughout the proofs of several of claims.
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Lemma B.1. Let fi...., fn R2 - a2 be set of functions such that for any , 2n R
Σ f((x)) =
(21)
Then fi,... are constant functions.
Proof. Without loss of generality, we will show for j [n] that fy is constant. Let 21,... In R
and : M/ Rd such that:
= Yo: Vi +
(22)
By equation 21
Σ f((1.) - Σ
f((y:) = O.
(23)
1=1
1=1
Using the condition on v, y results in:
f(()) Σ
f((v)) = E.(i)))
Efiuu + f5(2,) - f5(1i)
i=1
i=1
143
143
-E Si(zs) - E fi(xs) + 15(1z)-$$ss)
- fz(Uz)
(24)
its
it1
Therefore:
[5(,)) = {s(1)), Vaji E R.
(25)
Meaning fj is constant.
Lemma B.2. Let G < Sn acting on [n] and be the orbits of [n]² under the group
action ofG. G. Then for any nxxxd
Σ
A
A,.)((i,5) E Oxh.
(26)
where Il is the indicator function.
Proof. Each (i.j) belongs to exactly one orbit, therefore:
A Or) = Aijs
(27)
which concludes the proof.
B.2 PROOF FOR PROP.
Proof. We need to prove equivariant, i.e:
& = + ooo Vo E G.
(28)
For convenience, we will show the equivalent condition:
= - a $ oa, Va E G.
(29)
Let R be input vector. By definition, the output of the layer is given by:
(x),, Σ Pg.p (xp), Vq E [n].
(30)
Applying permutation G in the input space:
l(a x)g
Σ pa (2o 1(p)) Σ (g))
(31)
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where the last transition is valid since is bijection, and change the order of the summation.
Applying the inverse permutation on the output results
(o x))g - ( -x)o(())))) = Σ (1)))))))
(32)
The last transition uses equation 31 By the function sharing condition equation
(a ((ax)) po())) (z))
Σ
(xpp) = =b(x))
(33)
p=1
Therefore, & is equivariant.
B.3 PROOF FOR PROP.
Proof. To make the proof more concrete, we illustrate it using the case G Sn, and encourage the reader refer to Example 3 to see how the argument aligns with that specific case.
? Let O₁, O2,-..,. Oh denote the orbits of [n]] under the action of G, where H = [m]] / is the number of orbits. Let (gh- ph.) be representative of orbit O₁, for each h E [H].
In the example, we have H 2. The first orbit consists of the diagonal entries O2 = (ii, E
[n]], with (g), P1) (1, 1) as the representative. The second orbit includes all off-diagonal entries
O2 {(i. .) ú,j [n], i + j}. with P2) = (1, 2) as representative.
Since G-equivariant, we have:
o 1 xxx
(x) = 0, Vo E G.
(34)
Using Eq. equation 32 this condition simplifies to:
0 = (a T x))g - (xx),
n
n
(n()) Σ
(xp(op)
(35)
= Σ (z)) (zp)) VX. 1=1
Using Lemma B.1 each term in the sum must constant. Thus, functions with indices from the same orbit differ only by a constant. For any orbit On and for any (g.p) E On. we define:
C..p = =g &er.pn (.).
(36)
In the S,, case, it means that:
C2 (z) - (11(2); Ca.3 = Φ3,3(2) - (2,((2) :...
(37)
C2,3 22,2 (z) 21,((x) C1,3 = 01,3(2) - 21,((1) :...
We also define aj...., 0g as:
0g
4
C Cgpr g E [n].
(38)
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Substitute equation 36 in equation 35 results in:
(()) Pan (xp))
Σ (oo()) Pgn.pa (5p) Pen (Fp) (xp(sp))
Σ Cole)oop) Cg.
(39)
p=1
(o(o))(())
Σ
C..
p=1
Σ
Col
Σ
Ca.
p=1
p=1
ar(g)) ag
Where in the second last transition, we used the fact a a bijection, and we can change the order of the sum.
equation
39 suggests that:
0ig (a(()); Vo G.
(40)
In the example it means 01 02 as all the set [n] is in the same orbit.
Now, define & to be KA layer such that for any (q.p) On:
(. - 6g pa (-) + ap.
(41)
n
Since this definition depends only on orbit representative, & is G-equivariant FS-KA layer, i.e 6..2 (r()).a(p)) for all E G. To show that & and & represent the same function will use
Lemma B.2
2(x)g
Σ (zp)
p=1
De (stp (((.p))) E oN)
(42)
h=1
H
(on (xt) + ((( E Orh
p= 1 1=1
where I is the indicator function. Substitute equation 41 into the equation,
(Z)) aag + (Cgp))((lgpp)) E On}
p=14=1
Sa (1,))1(())))) On)
an I((q.p) +On)+
Cgpll((a,p)) Or).
n
p=1
p=1
(43)
We can simplify the first term using Lemma B.2
H
Sap (zp (x))))
O = Σ (pp( = 4(x)gg
(44)
p=1
1=1
Furthermore, we can use the lemma to simplify the third term:
Ca..ll(g.p) On Σ Cg. agq.
(45)
p=1
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Note that if (g. p) E On, there must exist permutation that maps @ to qn- Therefore by equation 40
it implies that 0g = gu- Then the second term can also be simplified as:
#
gg H(g.p) E EO,)
1
Σ
Tgg. 0O, = h=
0q = aq.
(46)
h=11
p=1
Combining it all, we get:
T(x)g x(x)g 04 + 0gg x(x)gg
(47)
Therefore, d and compute the same function, completing the proof.
B.4 PROOF FOR PROP. 3
Proof. The output of the layer given by:
((x) (x()))
(48)
Applying permutation G' in the input space:
Φ x) = Σ
Op(oo 2(p)) Σ
Oo(r) (Ip)
(49)
p=1
p=1
By the invariant FS condition (pr(p) = pp)
Φ x) = Σ do(p) (Tp) = Σ (xp)) 4xx)
(50)
p=1
Which concludes the proof.
B.5
PROOF FOR PROP. 4
Proof. Denote (bc)) R" R as a KA layer such that:
q (be) = p
(51)
i.e. broadcast n times.
& is G-invariant, then by the definition of (bhe):
T (o x)g 10(0 xx) -xxx,, Vo
(52)
On the other hand,
(a (lbc)( = (bb)) (x) ((() d(x)
(53)
Therefore, (be) is an G-equivariant layer. by Proposition 2 there exist G-equivariant FS-KA layer (bte) that is equivalent to ((be), Now, define &
((x) $ (x))
(54)
By the definition of (be), (((b))x:) (zr) Therefore, the layers are equivalent.
Using the equivariant FS condition for (fe) we can show that for any p [n] and G:
So()) = (1.((p) = - (1100)))))
(bc)
((be) ( = 11.p = Sp
((bc)
(55)
Therefore, $ is G-invariant FS-KA layer.
B.6 PROOF FOR PROP. 5
The proof of this proposition follows exactly the same structure as the proofs of Prop. Ir; vector of dimension di and Φy.p replaced with Φ9.P.
2 2 only with
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B.7 PROOF FOR PROP. 6
Proof. We begin by recalling Theorem 3.2 from Wang et al. (2024b) establishes that each layer of an MLP with activation function (+) maa((..)" can be represented by KAN with two hidden layers and grid size G = 2 with degree k B-splines, Here, & determines the polynomial degree of both the activation and the spline basis functions, while G specifies the number of grid intervals used in the spline parameterization. Their proof demonstrates this by representing the linear part of an MLP layer using a single KAN layer and the non-linear activation using another KAN layer with a diagonal structure. i,e for bounded domain Ω C RRdi and an affine layer L Rdd Rdout exists KA layers Idin - Rdout and Rdou + Rdowe such that:
((x) [(x); E
(56)
((x) = (x), Vx E ((2)
In the general case, the MLP has equivariant layers and invariant layer followed by standard MLP. As the work of Wang et al. (2024b) already handles standard MLPs, we will focus on showing that any equivariant or invariant linear layer with σ activation can be represented using one hidden layer FS- PS-KAN.
Let L Ω > paxdowi be equivariant affine layer. By Wang et al. (2024b) work exists a KA layer such that:
4 (x) = L(x), Vx EQ
(57)
i is equivariant but necessary FS layer. By Proposition existt FS-KA layer à such that:
① ((x) (x))
(58)
We will use ( defined on equation 56 to construct KA layer that simulate the ReLU activation:
T
V
q=p
(59)
q q
Note that V acts element-wise, thus S, S,,equivariant and therefore G-equivariant well. Further- more, V is a FS-KA layer.
Thus, T 6 is an FS-KAN with one hidden layer that simulates d o L on a bounded domain. The full network is realizable by simply composing the realizations of each layer while considering the previous layer's 's image to be the current layer's input domain.
B.8 PROOF FOR PROP. 7
Proof. We will prove the claim for a single equivariant layer &. The invariant layer proof follows the same proof structure. By definition, parameter-sharing equivariant layer can be written
((x)g Σ ((..)) (xp)
(60)
where each ((v.P) is function Rd R dus and the functions satisfy the equivariance FS condition:
T (g.p) o (+()))
Wo E G
(61)
Let > 0. By the universal approximation theorem Homik et al. (1989) for MLPs with ReLU activations, for each function p(9.p) (which is continuous on a compact domain), there exists two-layer MLP fla.)) with ReLU activations such that:
( - p(ap))
n
We assume all flu.p) have the same width dmaw by padding with zeros if needed. We denote w-(0.)) g((.)) as the weights and biases of the first input layer and w ((9.p) as those of the output linear layer.
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Because ((T-r) (()()))(())) may assume that flaz) f(r(a))(()) and therefore their param- eters are shared accordingly:
w|o0) ((o()))()) whop)) W2 w(((())()))
etc.
(62)
We now construct two-layer MLP approximating + ((x). the first layer L1 Raxdo R xdmux construction defined by:
(x)) (((r)) + (a.)
(63)
We will show this is indeed an equivariant linear map:
Li(a x).x w((p)) (a (n)) +b/ap)
(64)
On the other hand,
(a l (x)) (L1) W '(1).o 1()) (x0) + 404 1(1),0 1(0)
(65)
Using equation 62
(a. (x))) = w/(.p)
x la (o)) + ((.22) = [1 x)
(66)
L is equivariant linear map, it must be parameter-sharing linear layer (Wood & Shawe- Taylor 1996). Second layer construction L2 Rn xdmax + Ruddou will be defined as:
L2(x), = Σ (a p=1
W
Fq + bbo.0)
(67)
Similarly, L2 is also equivariant linear map, therefore has a parameter sharing structure. The param- eter sharing MLP f is given by:
f(x) = L (x))++
(68)
Where [+] the ReLU activation. By the construction of f:
f(x)g = Σ w
( (1laa) (()) + 1((.P) + b(a.)) Σ ((. (xp)
(69)
p=1
In total, the approximation error of the layer would be as:
[x(()) f(x)1 = Σ |1(x)) f(x)211
Σ1 Σ (g(e))(2p)) -(a.)) (zp))
Σ> Σ (2(())((2p)) f(ar)(z)))
(70)
n²
c/n²
Using Lemma 6 from Lim al. 2022) (Layer-wise universality implies universality), using com- position of the MLPs approximation for each layer approximates the whole MI.P up to any preci- sion.
B.9
PROOF FOR PROP.
g
Proof. Let P: : [n] x [m] + [nm]] be bijection. We define the flatten form of x E Ruxmxd as:
Veo(x); = Xx 1())
(71)
The group action of G' x H extend naturally as:
(g, P(i.j) P((g, h) (i, P(o(i),((i), GxH)) ((g, Vec(x)) = Veellg, h) xx)
(72)
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We will define the KA layer & on the flattened vector in implicit way:
@21-PI-9..22 = ( (g)((g)s))))Ppppp
(73)
On the one hand,
( .000
D (Tpr.
= pam1
Ri (mn. ( (xpr.)))
(74)
P1=1 p2=1
T ((eeea))))) p(mas))
p2
= क ((
On the other hand,
+ xz q n. Vec($(=))
(75)
Therefore,
((ee(x)) (ec(F(x))
(76)
Specifically, & is G x H equivariant, then by Prop. exists an equivalent equivariant FS-KA layer .. Note that G x H acts only on the first coordinate, and therefore, the use of our proposition is justified. By the FS condition:
J o0(1),66) No G
(77)
We will construct the equivalent KA layer &b for the tensor form by:
th क ((n.ai))P((ia
(78)
This construction implies too that ((Vec<z)) Vec( Tec((())) By the FS condition and by construc- tion of & we get:
fotti)a(tesMMG))MMaaa (P(66)))
P((id)))
(79)
For h = en (identity of H), we prove the sharing of the KA-sub layers, and for = eg. we prove the sharing within each sub-layer.
C IMPLEMENTATION DETAILS
implemented using the PyTorch framework and trained using the AdamW optimizer (Loshchilov & Hutter 2017). The FS- FS-KAN variants implementations are based on the Efficient KAN (Blealtan 2024) package and the ConvKAN (Bodner al. (024] GitHub repos- itory. with the exception of the symbolic formula experiments, which used the PYKAN library (Liu) al. 2024) and LBFGS optimizer for both KAN and FS-KAN implementations. For MLP-based models, we applied & weight regularization; for KAN-based models, we used the regularization provided by the Efficients KAN library. The exact parameter count of the invariant/equivariiant models are in Table 2 As for the non-invariant standard KAN in the point cloud classification task, the parameter counts vary between 42,240 and 403,040 depending on the specific configuration. The regularization term scaled by hyper-parameter in both cases. Models were trained on NVIDIA V100 GPUs. We trained the models with 5 different seeds all experiments and configu- rations.
All experiments
While Group Convolutional Neural Networks (GCNNs) Cohen Welling (016)) can also be con- sidered as baseline, they become computationally intractable for the large symmetry groups consider, as group convolutions scale with the size of the group. This illustrates how our approach more practical for real-world applications.
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Task
ES-KAN & Efficient ES-KAN
Sealed Baseline
Standard Baseline
Signal classification
33,834
35,467 (SSEM)
3,234,643 (SSEM)
cloud classification
55,584
55,976 (DeepSers)
55,159 (Point Transformer)
Recormeendation syslem
88,608
90,017 (DeepSets)
2.114,308 (DeepSets)
Table 2: Parameter count comparison across different tasks and methods
C.I LEARNING INVARIANT SYMBOLIC FORMULAS
The dataset generation follows the procedure provided in Liu et al. (2024). FS-KAN models consist of one equivariant layer with single feature channels followed by invariant layer. While the ES-KAN is illustrated for n = 3, it can handle varying input sizes. For the baseline KAN, we use one hidden layer of width 3. Both models use L-BFGS optimization with 25 steps and À 0.001. We refer to figure 7 for more results on different invariant formulas.
(a) Standard KAN
(b) FS-KAN
(c) Standard KAN
(d) FS-KAN
10s (oin(1zy)) sin((zz)) 3
Figure 7: Visualization of learned spline functions. Top row: [(x)
10s3 tnin(xay) (oin(we))
10x3 (annaaay/aaiisseee
e
Bottom row: f(x) = tanh(5(zf + x + x2gg - 1).
C.2 SIGNAL CLASSIFICATION EXPERIMENT
Data preparation. The dataset generation follows the procedure described on Maron et al. (2020) where each clean signal was randomly selected from predefined set of types sine, square, and sawtooth with T = 100 time steps. To construct set of noisy measurements, each signal is dupli- cated multiple times, adding independent Gaussian noise to each copy. All samples were generated using the code provided by Maron et al. (020)) Validation and test sets have fixed size of 300 samples each.
Networks and training. FS-KAN and efficient FS-KAN models have hidden layers of width 16, 16, 8, while the baseline DSS used hidden layers of width 160, 160, 80. Each layer was followed by batch norm well. We used the DSS implementation provided by Maron al. (020)_ As
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mentioned earlier, we also trained scaled DSS to match the number of parameters in the KAN-
based models. We used a batch size of 64 as in Maron et al. (020). For each configuration, we trained the models with 5 different seeds. We report in Table the training durations for all models.
Model
60
120
600
900
1200
DSS
76.1 + 0.3
93.9 = 0.6
240.0 + 0.3
356.7 + 25.9
481.0 1 30.8
scaled DSS
26.9 + 2.1
32.8 = 1.6
66.8 + 3.9
125.2 + 68.4
58.4 = 21.7
FS-KAN
422.6 + 2.5
499.7 + 5.1
1183.7 + 10.9
2025.4 29.4
2228.9
315.2
Efficient KS-
300.8 + 6.0
377.7 = 13.6
1146.5 + 275.4
1315.9 51.4
1654.8 E 194.8
(a) Low data regime
Model
6000
12000
18000
24000
30000
DSS
2c3 + 5.4
3.9c3 +: 7.6
5.9c3 1 8.5
7.8c3 1: 12.6
9.7c3 1 10.0
scaled DSS
150.3 1 1.8
385.4 147.6
1.0c3 1: 25.3
1.0c3 437.1
1.3c3 1 553.6
FS KAN
9.4e3 + 95.2
1.944 164.5
2.8e4 102.2
3.8e4 = 245.9
4.5e4 = 1098.8
Efficient FS-KAN
7.8c3 +: 79.8
1.5c4 4: 275.6
2.3e4 4: 219.2
3.1e4 +: 497.9
3.8c4 + 827.0
(b) High data regime
Table 3: Training time (seconds) of models for signal classification task 5 All models were trained for 200 epochs with batch size of 64.
Hyper-parameters. We perform a grid search over training learning rates / (10-2, 10-3 ..... 10-6) and regularization loss coefficients 7 E 10-1, 10-1) with 6000 training samples. peper-parameters are selected based on the highest validation accuracy. All models achieve their best validation performance with μ 10-3 and 17 10-
C.3 POINT CLOUDS CLASSIFICATION
Data preparation. For this experiment, we used ModelNer40 (Wu et al., 2015 which contains around 12 10 point clouds from different 40 classes. Each point cloud contains n 1024 points represented by d = 3 spatial coordinates. We used the normalized and sampled point clouds by Qi ctt al. (2017) with no data augmentations. For the validation set, we used a 90% 10% split of the full train set.
Networks and training. All invariant models are composed of equivariant layers, followed by an invariant layer and a standard output layer of width 40. ES-KAN and efficient KANs composed 2 equivariant FS-KA layers of width 36, each followed by a 1 1/ batch norm. The invariant layer has an output dimension of 36 well. The DeepSets model has the same architecture, only with width of 128 to match the number of parameters (5.6 x 10*). ). The non-equivariant KAN consists of hidden layers of width 16, but since cannot handle varying input sizes, its parameter count depends on the number of input points, ranging from 4.2 x 10a to 5.0 x 10a parameters. The transformer model uses an embedding dimension of 38 with layers (5.5 x 10* parameters), implemented with an input projection layer, transformer encoder layers with multi-head attention and a classification head with intermediate dimensionality reduction. For training and testing, we used batch size 32. Additionally, we conducted an experiment to measure the average runtime and peak GPU memory usage for both the training and inference steps for the FS-KANs and DecpSets, as reported in Tables 4 and 5 We also conducted experiments in the high-data regime using the full ModelNet40 dataset (8,859 training samples with 1,024 points per cloud), a standard benchmark in 3D computer vision. The results, presented in Table 6. demonstrate that FS-KAN variants achieve comparable accuracy with established architectures like DeepSets and Point- Transformer in this setting, complementing low-data regime analysis.
We performed grid search over training learning rates μ E {10-2 5 10-3, 10-3, 5 10-4, 10 x 10 ] and regularization loss coefficients 7) E {10 - 10-3 0}. The training and validation sets consist of point clouds with
1024 points. The training set size for the search 600. For FS-KAN models, the best validation performance is achieved with
10 and 7 10-5 while for DeepSets, the best configuration is μ 10-2 and 1) 0. The transformer model achieves optimal performance with = 10 - and 17 = 10 and the non- equivariant KAN with f 10-2 and =/
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(a) Train time per step (ms)
Model
n=64
n=
n=256
n=512
n=1024
DeepSets
4.09 = 0.46
3.43 + 1.13
4.22 + 0.65
2.53 + 1.02
1.97 0.24
Efficient FS-KAN
6.19 1.50
13.17 + 0.77
6.11 + 0.07
14.57 + 0.11
41.70 + 0.35
FS-KAN
6.33 + 0.14
7.21 2.26
7.53 + 1.62
61.78 + 0.53
Point- Transformer
6.76 0.43
6.56 + 0.09
6.60 + 0.24
17.67 - 6.76
63.41
20.48
Standard KANs
4.79 0.28
5.41 + 1.53
6.26 + 2.16
4.84 + 0.36
0.36
6.12 0.52
(h) Inference time per step (ms)
Model
n=64
n=128
n=256
n=512
n=1024
DeepSets
0.28 + 0.02
0.29 + 0.03
0.28 0.02
0.27 1 0.01
0.54 + 0.03
Efficient FS-KAN
2.00 + 0.04
2.00 + 0.06
2.00
0.03
4.94 + 0.03
13.39 0.09
FS-KAN
2.01 0.05
2.04 2.04 0.04
2.02 + 0.03
7.11 - 0.03
19.81 0.15
Point- Transformer
0.61 = 0.02
0.60 0.01
0.60 0.01
2.01 1.78
6.54 6.47
Standard KANs
1.20 = 0.01
1.20 0.01
1.22 0.00
1.21 + 0.01
1.19 0.01
Table 4: Comparison of training times and inference times (ms) per step for different models on the point cloud classification task (batch size 64) and for point clouds with points. We measure the runtime over 10 steps for each configuration and report the mean and standard deviation.
(a)
Training phase (MB)
Model
n=64
n=128
n=256
n=512
n=1024
DeepSets
31.23
45.28
73.37
129.56
241.94
Elficient FS-KAN
117.16
216.44
401.38
786.75
1556.00
PAS-KAN
157.81
301.25
566.66
1120.84
2222.47
Point-Transtonee Transformer
70.01
148.90
397.59
1323.53
4797.90
Standard KANs
18.78
21.16
25.99
35.46
54.53
(b) Inference phase (MB)
Model
n=64
n=128
n=256
n=512
n=1024
DeepSets
23.20
29.24
41.34
65.53
145.90
Efficient FS-KAN
43.05
68.36
1 114.71
215.14
406.52
FS-KAN
42.57
68.86
116.33
219.02
414.89
Soint-TTannformer
22.95
31.50
61.22
169.28
577.40
Standard KANs
18.89
21.35
26.31
36.07
55.70
Table 5: Comparison of GPU memory usage during training and inference (MB) for different models on the point cloud classification task and for point clouds with n points.
Model
DeepSets
FS-KAN
Efficient FS-KAN
Trant-Trrnse Standard KAN
Test Accuracy
0.844 E 0.005
0.837 0.007
0.840 + 0.005
0.840 0.008
0.232 0.074
Table 6: Test accuracy full ModelNet40 dataset (high-data regime)
C.4 CONTINUAL LEARNING ON POINT CLOUDS
Data preparation. For task A, followed the procedure described in Appx. For task B, applied geometric transformations to the task A dataset consisting of: (i) random scaling sampled uniformly from (0.95, 1.0), (ii) 3D rotation via axis-angle representation with rotation angles drawn from (0 0.52). and (iii) random translation with components sampled from (0, 0.12I).
C.3
Network and training. We the same models from Appx. C
We employ identical hyper- parameters for each model across both tasks. Each phase consisted of 1000 epochs, with the opti- mizer reinitialized at the beginning of each phase.
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Published as a conference paper at ICLR 2026
C.5 SEMI-SUPERVISER RATING PREDICTION EXPERIMENT
Data preparation. We used the MovieLens 100k dataset (Harper & Konstan 2015) along with smaller versions of Douban, Flixster, and Yahoo, introduced by Monti al. (2017). In the Flixster dataset, where ratings are given 0.5 increments, we labeled rating 0.5 as represented non-integer ratings as probabilistic mixtures (e.g.. rating of 3.5 was modeled as 50% chance of 3 and 50% chance of 4). For the Yahoo dataset, ratings ranging from 1 to 100 were clustered into 5 categories values.
and treated as
To simulate low-data scenarios, we uniformly sampled subsets of rows and columns from each rating matrix. Within each resulting sub-matrix, we further sub-sampled the observed entries to reduce training density. The train set was composed of uniformly sampled sub-matrices of the sparse matrix. The train set consists of 32 sub-matrices, and the test and validation sets have 64 samples each. The full data sampling process is illustrated in figure g Data split into train, validation, and test sets using the UI split defined by |aarper & Konstan (2015).
Network and training. All models are composed of 9 equivariant layers, each followed by skip connection and batch norm. KAN and FS-KAN architectures used layers width 16 while SSEM uses layers of width 256 detailed on Hartford et al. (2018). A smaller SSEM with width of 52 was also trained to match the parameter count of the KAN models. For each configuration, we trained the models with 5 different seeds.
hyper-parametee search We perform a grid search over training learning rates E {10-2, 2 10-3 3 10-4 10-5) and regularization loss coefficients 7) E {10-3, 10-5, 0}, using ap- proximately 600 ratings per run. blyper-arameeter are selected based on the lowest validation RMSE. The best-performing configurations are as follows:
KAN and Efficient FS-KAN: 10-4, .n 10-5 across all datasets.
* SSEM and Scaled SSEM:
- On ML-100K, Flixster, and Yahoo: = 10-4, 17 10-5.
-
On Douban:
* SSEM:
10-3, 10-5 * Scaled SSEM: μ = 10 10-3, 10-3,1=0 0.
D HYPERPARAMETER STUDY ON FS-KAN MODELS
We evaluated our FS-KAN model on point cloud classification (section 5 ablating over different architectural configurations. Specifically,
examined different widths (36, 64, 128), depths
L E {2, 4, 6]. and grid sizes {3, 4, 5, 6} of the spline functions. We note that increasing each of these parameters increases the model size. For each hypae-parrmettr configuration, we trained models on 600 samples, each containing n 256 points.
Results. Results are presented Table Our findings demonstrate that models with d consistently achieve superior test accuracies across all examined grid sizes, suggesting an optimal architectural configuration at moderate depth levels. Increasing the layer width from 36 to 128 yields notable performance improvements (approximately 0.010 1010-000 0.015), with width 128 achieving the best results. However, this comes at significant computational cost: as shown in Tables and 10 training time and memory usage scale super-linearly with width and more linearly with depth, while grid size increases show moderate impact. Regarding grid size, larger values (g = to g 6) generally provide better accuracy with diminishing returns. Considering both accuracy (Table 7 and computational efficiency (Tables 8 99 10). our results indicate that a configuration of 128, g 4 optimally balances classification performance and model complexity for point cloud classification tasks.
E LARGE LANGUAGE MODEL (LLM) USAGE
We used LLMs to assist with writing and polishing portions of this paper, including improving clarity of technical explanations, refining grammar and flow, and enhancing overall readability. All
26

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Published as a conference paper at ICLR 2026
Depth
Width
g =
9 =
9 ==
9 =
36
0.637 + 0.008
0.637 + 0.008
+.003
0.640 0.009
64
0.644 0.012
0.644 0.007
1.000
0.653
128
0.646 0.008
0.648 # 0.004
0.653 0.009
0.652 = 0.009
36
0.645
0.645 = 0.009
0.647 = 0.005
64
0.644 + 0.004
0.653 + 0.007
0.652 0.007
0.650 0.007
128
0.649 # 0.006 0.006
0.658 = 0.007
0.654 # 0.011
0.660 0.002
36
0.631 + 0.010
0.637 = 0.006
0.642 = 0.013
0.641 0.005
64
0.642 + 0.005
0.643 = 0.009
0.640 0.009
0.647 0.005
128
0.639 + 0.007
0.635 # 0.010
0.639 # 0.011
0.640 + 0.008
Table FS-KAN test accuracy results for different architectural configurations on point cloud clas-
sification
Table Average Training Time (ms) for Point Cloud Size n=256
Depth
Width
g
9
=
9 =
36
8.197 = 1.576
8.487 + 1.497
8.545 - 1.571
7.802 + 1.295
2
64
9.132 + 1.465
9.778 + 2.835
9.901 + 2.860
9.655 + 0.994
128
11.313 + 0.037
13.609 + 0.019
15.928 + 0.143
18.320 + 0.053
36
14.412 2.561
14.788 2.177
13.553 2.556
4
64
13.234 + 2.637
15.456 + 4.376
14.750 + 2.764
16.762 + 2.579
128
28.023 0.184
33.668 0.247
39.656 0.163
45.981 0.301
36
18.637 3.770
18.774 3.310
16.998 = 1.292
16.984 = 0.537
6
64
17.848 = 0.956
19.603 = 4.1990
19.048 = 0.050
23.211 = 0.104
128
44.329 0.257
54.041 0.241
63.376 0.232
73.487 + 0.292
Table 9: Average Inference Time (ms) for Point Cloud Size n=256
Depth
Width
=
= 4
g = 5
g = 6
36
2,002 = 0.024
2.039 + 0.019
2.002 + 0.034
2.177 + 0.298
2
1.989 0.033
2.249 0.549
2.375 = 0.801
2.057 + 0.128
128
3.821 + 0.013
4.853 + 0.004
5.916 + 0.003
7.040 + 0.004
36
3.449 = 0.257
3.461
3 0.090
3.442 = 0.076
4
64
3.348 = 0.083
3.536 4.536:0.0 0.482
3.487 + 0.417
3.425 + 0.196
128
8.066 = 0.096
10.448 0.063
13.149 0.073
15.946 0.113
4.808 = 0.225
4.640 0.064
4.700 = 0.094
4.697 = 0.117
6
64
4.630 = 0.054
4.879 + 0.303
5,040 + 0.223
5.574 + 0.144
128
12.838 0.022
16.699 0.012
20.867 0.037
25.038 0.028
research contributions, experimental design, analysis, and conclusions are entirely our own work. The LLM was used solely as a writing assistance tool to improve quality.
27

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Published as a conference paper at ICLR 2026
Table 10: Peak Training GPU Memory Usage (MB) for Point Cloud Size n=256
Depth
Width
g =
3
g =
9
=
36
234.305
269.117
301.245
330.756
64
399.340
453.631
504.164
556.456
128
781.065
885.992
990.793
1096.722
36
498.823
575.209
646.656
710.748
4
64
865.529
984.267
1099.116
1215.730
128
1718.178
1952.368
2186.433
2421.625
36
763.967
882.019
994.067
1091.333
6
64
1331.842
1514.778
1694.068
1875.005
128
2655.290
3018.744
3382.073
3746.528
Table 11: Peak Inference GPU Memory Usage (MB) for Point Cloud Size n=256
Depth
Width
=
g y=1
g =
36
58.441
63.805
68.857
72.668
64
89.884
98.951
106.262
114.455
128
166.093
181.824
198.429
215.912
36
59.138
63,706
69.743
72.734
64
91.905
101.224
108.784
117.229
128
174.123
190.856
208.463
226.948
36
59.771
64.389
70.504
73.643
64
93.925
103.496
111.307
120.004
128
182.153
199.888
218.498
237.984
NoMe5 Rating Mabrix
Train Mask
Teat Mask
Sampling
columns uniformly
Removing training tabels
Sampling and columns uniformly
Figure 8: Rating matrix sub-sampling pipeline. We uniformly sample users and items, followed by removing observed entries to simulate extreme data scarcity. Finally, repeatedly sample rows and columns to generate multiple sub-matrices used as training and test batches.
28


Reviews:
[{"review_id": "QLtIomx0yd", "reviewer": "Reviewer_5F3P", "summary": "The paper proposes a permutation-equivariant neural network based on the Kolmogorov–Arnold framework.", "questions": "Could the authors provide additional evidence or comparisons regarding runtime and memory efficiency with other baselines? Currently, the analysis only includes DeepSets, making it difficult to assess how FS-KAN performs relative to a broader range of architectures.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is well written and self-contained. Even for readers who are not deeply familiar with KANs, the main ideas are clear and accessible.\n\n- The construction of the proposed permutation-equivariant and permutation-invariant KANs is clearly presented. It is supported by solid theoretical guarantees, including expressivity results (Propositions 2 and 4) and expressive power analysis (Section 4). The accompanying visualizations effectively illustrate the concepts, and to the best of my knowledge, this formulation is novel within the context of permutation-equivariant networks.\n\n- The experimental section covers a diverse set of symmetry-structured tasks, such as signal classification, point-cloud classification, and semi-supervised rating prediction, demonstrating the versatility of the proposed framework.", "weaknesses": "- While the theoretical results are clearly presented and technically sound, they are somewhat expected and incremental. Propositions 2 and 4 follow naturally from basic algebraic reasoning, and the expressive power result largely derives from existing results on standard KANs.\n\n\n- I find the motivation from the KAN literature somewhat confusing. KANs are often considered computationally heavy and inefficient compared to mainstream architectures such as MLPs, CNNs, or Transformers. Although spline-based parameterizations can alleviate some of these issues, KAN-based models are still known for their slower training and higher memory consumption. While the paper provides several theoretical guarantees on expressivity, I believe the empirical performance and efficiency aspects are more crucial in this context. **The trade-off between accuracy and computational cost remains too large**. This makes the practical necessity of applying KANs to permutation-equivariant networks questionable. Although the authors introduce an efficient variant of FS-KAN to mitigate these issues, the improvement is modest, and in some cases, the efficient version performs comparably or even worse than the full FS-KAN model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FdtOoukG91", "reviewer": "Reviewer_4Zk8", "summary": "The paper introduces FS-KAN, a principled framework for building permutation-equivariant and invariant KAN layers by sharing functions (rather than weights) across positions. The authors prove that FS-KANs have expressive power equivalent (on bounded domains) to parameter-sharing MLPs; this lets known universality and expressivity results for equivariant networks (e.g., DeepSets/GNN variants) transfer directly. Empirically, across point-cloud classification (ModelNet40, including a continual-learning protocol) and recommender datasets with $S_n \\times S_m$ symmetry, FS-KANs show clear gains in low-data regimes and reduced forgetting, while preserving KAN interpretability. The paper also presents an “efficient” FS-KAN variant that reduces (but does not eliminate) runtime/memory overheads. Overall, the work unifies scattered equivariant-KAN designs and offers a clean blueprint to apply KA layers under arbitrary permutation groups.", "questions": "- What is the impact of the number/order of knots, the function parameterization (splines vs. alternatives), and the degree of sharing (full vs. partial) on performance and computational cost?\n\n- How would the theoretical results change if the permutation group were replaced by a continuous group (e.g., rotations)?\n\n- Is there an intuitive explanation for why FS-KANs can exactly represent any parameter-sharing MLP, whereas parameter-sharing MLPs can only approximate, but not exactly represent, arbitrary FS-KANs?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper presents an elegant generalization of parameter sharing to function sharing in KANs; the framework subsumes prior set/image KANs and handles arbitrary permutation groups and multi-channel tensors.\n\n- The authors proves interesting bidirectional results: on the one hand, FS-KAN can be used to represent ant parameter-sharing MLPs; on the other hand, parameter-sharing MLPs can be used to approximate any FS-KANs. These results clarify exact expressivity and transfer known universality/limits from parameter-sharing MLPs to to FS-KANs.\n\n- On low-data point-cloud and recommendation settings, FS-KANs deliver consistent improvements in accuracy/RMSE and lower forgetting in continual learning.", "weaknesses": "My main concern with this paper lies in the weaknesses of the experimental evaluation.\n\n- Regarding compute/latency, even the efficient variant is ~1.5× faster than vanilla FS-KAN yet still slower than DeepSets, and its memory usage remains higher.\n\n- Figure 5 and Table 1 show that FS-KAN’s improvements over the baselines are marginal; if anything, they suggest parity with older baselines such as Point Transformer.\n\n- No large-scale benchmarks are discussed. Given the high computational overhead and only marginal gains over DeepSets, it is plausible that DeepSets may slightly surpass FS-KAN at larger training sizes.\n\nHowever, because the theoretical results constitute the paper’s main contribution, and are both interesting and valuable to the ML community, I assign a positive score at this time.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iM1zoZYGlh", "reviewer": "Reviewer_9HVo", "summary": "The paper introduces a new approach to realize equivariant (or invariant) KANs. The underlying mechanism is a direct generalization of the weight-sharing used for MLPs, where now it becomes a function sharing of the learnable KAN functional components.\nThe work claims that the approach generalizes existing equivariant KAN architectures, and that the function-sharing approach is general, since any equivariant KA layer can be exactly represented by a FS-KA layer.\nThe authors then introduce an efficient version of the layer, provide examples intended to demonstrate how FS-KAN can be designed and how they generalize existing architectures, and prove that MLPs and FS-KANs can approximate each other to arbitrary accuracy. This fact is then used to derive universality properties of FS-KANs.\nThe experimental section convincingly demonstrate the potential of the new architecture.", "questions": "Apart from the points discussed above, there are the following minor points: \n \n- The $\\star$ notation in (1) is standard in KANs, but should be briefly introduced.\n\n- Paragraph \"Invariance, equivariance, and standard parameter-sharing\": The problem is formulated for $V=V'=R^n$, instead of general $V=R^n$, $V'=R^{n'}$. It is unclear at this stage if this is a necessary assumption for the definition of FS-KA layers, or if the assumption is just for the sake of simplifying the presentation. Please clarify this.\n\n- Equation (3) gives the wrong impression that only constant, zero-sum offsets are allowed. While this is clear from the context, I suggest modifying the example in (3) by using a functional offset.\n\n- Proof of Proposition 6: The notions of grid size and $B$-splines (\"grid size $G = 2$ with degree $k$ B-splines\") are nowhere defined before.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The related work clearly highlights the motivation of the work (simplification of existing approaches, variable size inputs, general permutation groups).\n- The topic of combining KANs with equivariant modelling is of current interest, and the results cover both efficient implementation and analytical properties.\n- The experimental section is clearly structured and it addresses a clearly defined set of questions.", "weaknesses": "- Efficient FS-KA layers: This seems to be a central novelty of the paper, as it enables applications and scaling to realistic problems. However, their definition is mainly textual and hard to process. I suggest to revise the paragraph \"Efficient FS-KA layers\" and make their definition more explicit.\n\n- Generalization of the existing literature: There is an emphasis in Sections 1-2 on the fact that FS-KANs generalize and unify existing approaches. This fact is however not sufficiently demonstrated in the paper. Section 3.3 in particular has a \"FS-KANs generalize previous equivariant KANs\" paragraph, which only mentions some similarity with the existing literature. \n\n- Theoretical analysis: Proposition 7 is formulated in an imprecise manner, as there is no dependency on the target accuracy $\\varepsilon$, while clearly the MLP $f$ is in fact $f(\\varepsilon)$. This becomes evident by inspecting the proof, but it should made clear upfront. \nEspecially, the width $d_{\\max}$ of the MLP $f$ seems to be unboundedly growing in $\\varepsilon$, since the construction is based on the 1989 Hornik et al. result. \nApart from updating the statement of Proposition 7, I also recommend to update the following sentences:  (I) \"Both proofs provide constructive methods for building the corresponding networks\",  following Proposition 7, which is not correct as it stands; (ii) \"we show that for a specific permutation group G, both model families have the same expressive power in the uniform approximation sense\", should rather state the the two families are dense in the same set of functions.\n\n- The experimental results compare FS-KANs only with MLPs with shared parameters. Comparisons with other equivariant architectures should be considered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lO64NhLIPp", "reviewer": "Reviewer_mRDw", "summary": "This paper introduces FS-KAN, a principled framework for building permutation-equivariant and invariant neural networks by extending parameter-sharing schemes to function sharing. The approach unifies and generalizes prior work on equivariant KANs, providing a theoretical proof that FS-KANs have equivalent expressive power to traditional parameter-sharing MLPs. Empirically, FS-KANs achieve strong performance and data efficiency across diverse symmetry-aware tasks, including signal classification, point cloud recognition, and matrix completion, particularly in low-data regimes. The architecture retains the interpretability and adaptability of KANs while significantly improving generalization for data with built-in symmetries", "questions": "- How does FS-KAN training scale computationally and memory-wise with input dimension and group size, compared to DeepSets or GNN baselines?\n- Can FS-KANs be adapted for continuous or mixed-group symmetries (e.g., rotations, reflections) beyond discrete permutations?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper presents a theoretically sound and general framework for permutation-equivariant KANs, extending the idea of parameter sharing to function-level sharing. It also provides clear mathematical proofs showing expressivity equivalence with parameter-sharing MLPs.\n- The proposed architecture maintains the interpretability of KANs, as demonstrated in visual examples where symmetric function structures emerge naturally.\n- The efficient FS-KAN variant reduces computational and memory costs while preserving equivariance, making the method more practical.", "weaknesses": "- Despite the good performance, computational cost remains high, particularly compared to standard MLP-based models in high-data regimes.\n- The empirical analysis, while broad, relies mostly on synthetic or small-scale datasets, leaving scalability to large real-world systems untested.\n- The practical interpretability claims are qualitative; the paper lacks systematic metrics quantifying how function sharing aids transparency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "s1SV8kPCEF", "reviewer": "Reviewer_DxdS", "summary": "The paper proposes FS-KAN and function sharing, enabling Kolmogorov–Arnold Networks (KAN) to implement group equivariance. This design unifies parameter sharing and functions sharing enabling transfer of known results (e.g., universality). Experiments show competitive performance, especially in low-data regimes.", "questions": "**Interpretability claim.**  \nOne of KANs’ most advertised features is improved interpretability, and FS-KANs build on this. However, this claim is contested: learnable activations are not identifiable up to scale and permutations, which makes post-hoc interpretation fragile at the local scale. At larger scales, even the KAN authors acknowledge that “a KAN may only remain interpretable when the network scale is relatively small” (p. 21 of [3]). \nTo what extent does enforcing equivariance mitigate this fragility?\n\n[3] Z. Liu, \"KAN 2.0: Kolmogorov-Arnold Networks Meet Science\", 2025", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "- Well written and clearly structured with working examples.\n- Proposition 2 is a rigorous result which requires new and different proofs from the ones known in the weight sharing literature for equivariant neural networks. \n- The authors propose a perspective that unifies neural networks and KANs. Function sharing for KANs cleanly recovers common equivariant designs for permutation representations of groups.", "weaknesses": "- **Insufficient novelty.** The introduction of the concept of function sharing is a simple adaptation of the regular concept of weight sharing for equivariant neural networks, as presented in [1]. The majority of main theoretical statements except for Proposition 2 largely transfer as it is from the weight sharing literature for equivariant deep learning.\n- **Missing analysis of low-frequency learning and spectral bias.** This topic is currently central to KAN research [2]. In particular, KANs are known to avoid the low-frequency bias of standard neural networks, enabling them to perform better on high-frequency targets (e.g., certain PDE solutions). These properties make KANs especially relevant for AI-for-Science; yet the manuscript does not investigate this direction, even though enforcing equivariance to the symmetries of specific PDEs could potentially improve performance.\n\n[1] S. Ravanbakhsh, *Equivariance Through Parameter-Sharing*, 2017 \\\n[2] Y. Wang et al., *On the expressiveness and spectral bias of KANs*, 2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IkuQgcSKvC", "reviewer": "Reviewer_G8oo", "summary": "This paper proposes FS-KAN, a framework that enforces permutation equivariance and invariance in Kolmogorov–Arnold Networks by sharing functions rather than weights. The authors establish an expressivity equivalence with parameter-sharing ReLU MLPs, showing that FS-KANs match their expressivity and thus inherit known universality guarantees for symmetry aware models. Empirically, across multiple symmetry tasks, FS-KANs outperform other baselines in low-data regimes, delivering better data efficiency while retaining KANs interpretability.", "questions": "- Does function sharing over-constrain FS-KAN? With many spline functions tied across positions (Appendix Fig. 7), could the sharing pattern overly limit capacity and hinder learning in practice?\n- The appendix indicates substantially higher runtime and memory than other baselines. Can runtime/memory for FS-KAN be improved?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper proves that FS-KAN captures all G-equivariant/invariant KA layers via function sharing. The proof is easy to follow with standard techniques in equivariant/invariant layer design.\n- Emperical evident suggests that their FS-KAN is better comparing to other baselines in low data regime.", "weaknesses": "- In the point-cloud task (e.g., at ~1k training samples), other baseline match or surpass FS-KAN, suggesting its benefit diminishes as training size grows. This reduces the practical appeal of FS-KAN beyond low data settings.\n- The computational runtime/memory analysis from Appendix C shows that FS-KAN incurs substantially higher runtime and memory usage than the other baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)