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

2025-09-29

AI Summary

Existing methods lack systematic modeling capabilities for general permutation symmetry groups (e.g., $S_n$ and its subgroups). Method: We propose FS-KAN—the first unified equivariant and invariant Kolmogorov–Arnold network framework tailored to arbitrary permutation symmetry groups. Grounded in the Kolmogorov–Arnold representation theorem and group representation theory, FS-KAN employs learnable piecewise spline functions to enforce exact functional sharing, thereby rigorously guaranteeing equivariance or invariance under the target group action. Its expressive power is provably equivalent to conventional parameter-sharing networks. Contribution/Results: We theoretically establish FS-KAN’s strong interpretability and superior low-sample generalization. Experiments demonstrate significant improvements in data efficiency across diverse symmetric data domains—including point clouds, graphs, and sets—with particularly notable performance gains in low-resource regimes—while preserving architectural flexibility and intrinsic interpretability.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Developing permutation equivariant KANs for arbitrary symmetry groups
• Unifying parameter-sharing schemes with Kolmogorov-Arnold network architecture
• Enhancing data efficiency while preserving KAN interpretability and adaptability

Innovation

• FS-KAN uses function sharing for permutation symmetry
• Generalizes parameter-sharing to Kolmogorov-Arnold networks
• Provides theoretical guarantees matching standard network expressivity