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PolyKAN: A Polyhedral Analysis Framework for Provable and Approximately Optimal KAN Compression

Di Zhang

TL;DR

PolyKAN addresses the challenge of compressing Kolmogorov-Arnol d Networks (KANs) with formal guarantees on size and approximation error. It introduces a complete polyhedral characterization of KANs, along with a formal theory of $\epsilon$-equivalent compression via region merging and a rigorous error propagation framework across layers. A dynamic-programming algorithm achieves provable optimal compression for univariate splines and offers approximately optimal solutions for multilayer KANs with polynomial-time guarantees. This framework provides a principled, verifiable path to deploying interpretable spline-based networks at scale, and points to future work on tighter bounds and extensions to other spline families.

Abstract

Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to traditional Multi-Layer Perceptrons (MLPs), offering enhanced interpretability and a solid mathematical foundation. However, their parameter efficiency remains a significant challenge for practical deployment. This paper introduces PolyKAN, a novel theoretical framework for KAN compression that provides formal guarantees on both model size reduction and approximation error. By leveraging the inherent piecewise polynomial structure of KANs, we formulate the compression problem as a polyhedral region merging task. We establish a rigorous polyhedral characterization of KANs, develop a complete theory of $ε$-equivalent compression, and design a dynamic programming algorithm that achieves approximately optimal compression under specified error bounds. Our theoretical analysis demonstrates that PolyKAN achieves provably near-optimal compression while maintaining strict error control, with guaranteed global optimality for univariate spline functions. This framework provides the first formal foundation for KAN compression with mathematical guarantees, opening new directions for the efficient deployment of interpretable neural architectures.

PolyKAN: A Polyhedral Analysis Framework for Provable and Approximately Optimal KAN Compression

TL;DR

PolyKAN addresses the challenge of compressing Kolmogorov-Arnol d Networks (KANs) with formal guarantees on size and approximation error. It introduces a complete polyhedral characterization of KANs, along with a formal theory of -equivalent compression via region merging and a rigorous error propagation framework across layers. A dynamic-programming algorithm achieves provable optimal compression for univariate splines and offers approximately optimal solutions for multilayer KANs with polynomial-time guarantees. This framework provides a principled, verifiable path to deploying interpretable spline-based networks at scale, and points to future work on tighter bounds and extensions to other spline families.

Abstract

Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to traditional Multi-Layer Perceptrons (MLPs), offering enhanced interpretability and a solid mathematical foundation. However, their parameter efficiency remains a significant challenge for practical deployment. This paper introduces PolyKAN, a novel theoretical framework for KAN compression that provides formal guarantees on both model size reduction and approximation error. By leveraging the inherent piecewise polynomial structure of KANs, we formulate the compression problem as a polyhedral region merging task. We establish a rigorous polyhedral characterization of KANs, develop a complete theory of -equivalent compression, and design a dynamic programming algorithm that achieves approximately optimal compression under specified error bounds. Our theoretical analysis demonstrates that PolyKAN achieves provably near-optimal compression while maintaining strict error control, with guaranteed global optimality for univariate spline functions. This framework provides the first formal foundation for KAN compression with mathematical guarantees, opening new directions for the efficient deployment of interpretable neural architectures.

Paper Structure

This paper contains 19 sections, 11 theorems, 7 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Kolmogorov-Arnold Networks and Spline Theory
  4. Neural Network Compression
  5. Polyhedral Theory in Deep Learning
  6. Polyhedral Characterization of KANs
  7. Univariate Spline Partitions
  8. Multilayer KAN Partitions
  9. Structural Properties of KAN Polyhedra
  10. Theory of Provable KAN Compression
  11. Formal Problem Statement
  12. Computational Complexity Analysis
  13. Theory of Region Mergability
  14. Error Propagation Analysis
  15. Approximately Optimal Compression Algorithms
  16. ...and 4 more sections

Key Result

Theorem 1

The input space partition of an $L$-layer KAN is a refinement of the partitions induced by spline functions across all layers. The total number of linear regions satisfies: where $k^{(l)}_{j,i}$ is the number of knots in the spline function $s^{(l)}_{j,i}$ connecting node $i$ in layer $l-1$ to node $j$ in layer $l$.

Theorems & Definitions (24)

  • Definition 1: Spline Polyhedron
  • Theorem 1: KAN Polyhedral Region Structure
  • proof
  • Lemma 1: Axis-Alignedness
  • proof
  • Lemma 2: Rectangular Structure
  • proof
  • Lemma 3: Function Regularity
  • Definition 2: $\epsilon$-Equivalent Compression
  • Theorem 2: NP-Hardness
  • ...and 14 more