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FS-KAN: Permutation Equivariant Kolmogorov-Arnold Networks via Function Sharing

Ran Elbaz, Guy Bar-Shalom, Yam Eitan, Fabrizio Frasca, Haggai Maron

TL;DR

FS-KAN addresses learning with permutation symmetries by introducing a principled framework that builds invariant and equivariant Kolmogorov–Arnold Network layers through function sharing. It proves that FS-KANs match the expressivity of standard parameter-sharing MLPs, enabling transfer of universal approximation results to FS-KANs, and demonstrates improved data efficiency in low-data regimes across tasks such as signal classification, point-cloud classification, and matrix completion. The approach generalizes to direct-product and high-order tensor symmetries and preserves interpretability, offering a practical, symmetry-aware alternative to traditional MLPs and graph neural networks. Overall, FS-KAN unifies prior equivariant KA approaches under a single theory, with empirical benefits and clear avenues for speeding up computation in future work.

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.

FS-KAN: Permutation Equivariant Kolmogorov-Arnold Networks via Function Sharing

TL;DR

FS-KAN addresses learning with permutation symmetries by introducing a principled framework that builds invariant and equivariant Kolmogorov–Arnold Network layers through function sharing. It proves that FS-KANs match the expressivity of standard parameter-sharing MLPs, enabling transfer of universal approximation results to FS-KANs, and demonstrates improved data efficiency in low-data regimes across tasks such as signal classification, point-cloud classification, and matrix completion. The approach generalizes to direct-product and high-order tensor symmetries and preserves interpretability, offering a practical, symmetry-aware alternative to traditional MLPs and graph neural networks. Overall, FS-KAN unifies prior equivariant KA approaches under a single theory, with empirical benefits and clear avenues for speeding up computation in future work.

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.

Paper Structure

This paper contains 36 sections, 11 theorems, 80 equations, 8 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. invariant and Equivariant function sharing Networks
  4. Preliminaries
  5. Invariant and equivariant Function Sharing KA layers
  6. Examples and important cases
  7. A theoretical analysis of the expressive power of FS-KANs
  8. Experiments
  9. Conclusion
  10. Acknowledgments
  11. Generalizations of FS-KA layers
  12. Generalization to direct product symmetries
  13. Higher-order tensors
  14. Efficient FS-KA layers
  15. Examples.
  16. ...and 21 more sections

Key Result

Proposition 1

Let $\Phi$ be a KA layer with $n_\text{in}=n_\text{out}=n$. We say that $\Phi$ is a G-equivariant Function Sharing (FS) KA layer if and only if Moreover, any such G-equivariant FS-KA layer is G-equivariant (proof in Appx. appendix: proof of equivariant fs layer).

Figures (8)

  • Figure 1: Parameter-sharing \ref{['subfig:ws for c5']} and FS \ref{['subfig:fs for c5']} for $C_5$ equivariant layers (1D-convolutional layers). Function sharing constrains the functions to be shared across the matrix according to the group action.
  • Figure 2: Equivariant FS-KA layers for different groups. \ref{['subfig:fs for Sn']} is $S_3$-equivariant (with $d=4$ feature channels), \ref{['subfig:fs for sn x cm']} is $S_3 \times C_5$-equivariant ($d=1$) and \ref{['subfig:fs for sn x sm']} is $S_3\times S_4$ equivariant ($d=1$). The functions in each sub-layer in \ref{['subfig:fs for sn x cm']} are shared across the generalized diagonals (internal sharing w.r.t $C_4$), while the sub-layers themselves are shared according to equation \ref{['eq: deepset formulation']} (external sharing w.r.t $S_3$).
  • Figure 3: Visualization of learned spline functions for learning $f(x) = \exp(-x_1^2-x_2^2-x_3^2)$. FS-KAN \ref{['fig:fskan_viz']} 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 interpretability.
  • Figure 4: Test accuracy for signal classification with multiple measurements with varying train 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.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Corollary 4.1
  • Proposition 8
  • Lemma B.1
  • ...and 10 more