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Clifford Kolmogorov-Arnold Networks

Matthias Wolff, Francesco Alesiani, Christof Duhme, Xiaoyi Jiang

TL;DR

Clifford Kolmogorov-Arnold Network (ClKAN) extends Kolmogorov-Arnold networks to Clifford algebras for high-dimensional, hypercomplex function approximation. It introduces Clifford-aware radial basis functions and two grid strategies, including a Randomized Quasi Monte Carlo (RQMC) Sobol grid, to mitigate exponential growth in parameters and improve coverage of the Clifford space. The paper establishes expressivity results for Sobol-CliffordKAN and demonstrates competitive performance against complex-valued baselines on synthetic formulas, holography data, and higher-dimensional Clifford algebras, while achieving significant parameter reductions in many high-dimensional settings. It also analyzes batch normalization variants tailored to Clifford domains and provides open-source code to promote reproducibility and further research. The findings suggest ClKAN enables scalable, interpretable models for physics-inspired and engineering tasks that naturally live in geometric algebras.

Abstract

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford algebra spaces. We propose the use of Randomized Quasi Monte Carlo grid generation as a solution to the exponential scaling associated with higher dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics inspired tasks.

Clifford Kolmogorov-Arnold Networks

TL;DR

Clifford Kolmogorov-Arnold Network (ClKAN) extends Kolmogorov-Arnold networks to Clifford algebras for high-dimensional, hypercomplex function approximation. It introduces Clifford-aware radial basis functions and two grid strategies, including a Randomized Quasi Monte Carlo (RQMC) Sobol grid, to mitigate exponential growth in parameters and improve coverage of the Clifford space. The paper establishes expressivity results for Sobol-CliffordKAN and demonstrates competitive performance against complex-valued baselines on synthetic formulas, holography data, and higher-dimensional Clifford algebras, while achieving significant parameter reductions in many high-dimensional settings. It also analyzes batch normalization variants tailored to Clifford domains and provides open-source code to promote reproducibility and further research. The findings suggest ClKAN enables scalable, interpretable models for physics-inspired and engineering tasks that naturally live in geometric algebras.

Abstract

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford algebra spaces. We propose the use of Randomized Quasi Monte Carlo grid generation as a solution to the exponential scaling associated with higher dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics inspired tasks.
Paper Structure (24 sections, 1 theorem, 11 equations, 5 figures, 4 tables)

This paper contains 24 sections, 1 theorem, 11 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Kolmogorov-Arnold Function Representation
  4. Clifford Algebra
  5. Background
  6. Clifford algebra
  7. Complex-valued Kolmogorov-Arnold Network
  8. Methodology
  9. ClKAN Radial Basis Functions
  10. ClKAN uniform grid
  11. RQMC Sobol Grid
  12. Expressivity of Sobol-CliffordKAN
  13. Batch normalization
  14. Experiments
  15. Complex-valued basic synthetic formulas
  16. ...and 9 more sections

Key Result

Lemma 4.2

If we use the Sobol grid of size $n$, then $\Phi(x) = \sum_{g \in G} w_g~\phi(x-g)$ is an unbiased estimator of $h(x) = \int_{[0,1]^d} g(y) \phi(x-y) dy$ with variance $O(n^{-1})$.

Figures (5)

  • Figure 1: Visualization of $\mathbb{C}l(3)$ grades (scalars, vectors, bivectors and trivectors).
  • Figure 2: Comparison of full grid, random grid, and quasi-random Sobol grid using Sobol sequences with 64 grid points each.
  • Figure 3: Visualization of dimension-wise (orange), node-wise (green) and component-wise (blue) batch normalization.
  • Figure 4: Overview of mse for all experiments for complex-valued synthetic function fitting tasks on formulas \ref{['eq:syntheticFormulasSquare']} and \ref{['eq:syntheticFormulasSquaresquare']} with color indicating the type of batch normalization, shape the type of rbf and shading the architecture size with shaded regions representing the small model architecture. X-axis labels correspond to cvkan baseline, Full grid and Sobol grid with number of grid points per dimension. For better readability S-5 and S-7 have been omitted. Y-axis shown in log-scale.
  • Figure 5: Function-fitting experiments on higher dimensional Clifford algebras. Color represents Clifford algebra used, shape the dataset mult\ref{['eq:syntheticFormulasMult']}, square\ref{['eq:syntheticFormulasSquare']} and squaresquare\ref{['eq:syntheticFormulasSquaresquare']} and shading the model architecture per dataset with shaded regions representing the small model architecture. X-axis labels correspond to Full grid and Sobol grid with number of grid points per dimension. For better readability S-5 and S-7 have been omitted. Y-axis shown in log scale.

Theorems & Definitions (1)

  • Lemma 4.2