Clifford-KAN
Mathematical Foundations of Clifford-KANs
This technical note details the formal integration of Clifford Algebra $\mathcal{C}\ell_{p,q}$ into the Kolmogorov-Arnold Network (KAN) architecture.
1. The Kolmogorov-Arnold Representation Theorem
The CKAN architecture is rooted in the Kolmogorov-Arnold Representation Theorem, which states that any multivariate continuous function $f: [0,1]^n \to \mathbb{R}$ can be represented as:
$$f(x_1, \dots, x_n) = \sum_{q=1}^{2n+1} \Phi_q \left( \sum_{p=1}^n \phi_{q,p}(x_p) \right)$$Where:
- $\phi_{q,p}$ are $n(2n+1)$ univariate continuous functions (inner functions).
- $\Phi_q$ are $2n+1$ univariate continuous functions (outer functions).
In a KAN, these functions are parameterized as learnable B-splines.
2. Clifford Algebra $\mathcal{C}\ell_{p,q,r}$
A Clifford Algebra is an associative algebra generated by a vector space $V$ with a quadratic form $Q$. For a basis $\{e_1, e_2, \dots, e_d\}$, the fundamental geometric product is governed by the relation:
$$e_i e_j + e_j e_i = 2 \eta_{ij} \mathbb{1}$$Where $\eta_{ij}$ is the metric tensor of signature $(p, q,r)$. A general multivector $u \in \mathcal{C}\ell_{p,q,r}$ is a linear combination of $2^d$ basis elements:
$$u = \underbrace{u_{\emptyset}}_{\text{scalar}} + \underbrace{\sum u_i e_i}_{\text{vectors}} + \underbrace{\sum u_{ij} e_{ij}}_{\text{bivectors}} + \dots + \underbrace{u_{12\dots d} e_{12\dots d}}_{\text{pseudoscalar}}$$3. The Clifford-KAN Formulation
The CKAN extends the scalar KAN by allowing the learnable functions to operate on multivectors. There are two primary ways this is mathematically achieved in the paper:
A. Component-wise (Grade-Specific) Splines
The multivector $u$ is decomposed into its $2^d$ scalar components $u_A$. The edge transformation $\Psi$ is defined as:
$$\Psi(u) = \sum_{A \in \mathcal{I}} \phi_A(u_A) \mathbf{e}_A$$where $\phi_A$ are independent learnable univariate B-splines for each basis blade $\mathbf{e}_A$. This ensures that the network can learn different behaviors for different geometric grades (e.g., treating scalars differently from bivectors).
B. Clifford-Valued Functions
For a more coupled approach, the univariate function is extended to the Clifford domain using a series expansion or a grade-preserving mapping:
$$\Phi(u) = \sum_{k=0}^K w_k u^k$$where $u^k$ is computed using the Clifford geometric product, allowing for non-linear interactions between different blades (e.g., a vector squared becoming a scalar).
4. Forward Propagation in CKAN Layers
For a layer with $N_{in}$ input multivectors and $N_{out}$ output multivectors, the $j$-th output multivector $u_{out,j}$ is calculated as:
$$u_{out,j} = \text{Act} \left( \sum_{i=1}^{N_{in}} \Psi_{i,j}(u_{in,i}) \right)$$Where:
- $\Psi_{i,j}$ is the learnable Clifford-valued spline function on the edge.
- The summation is performed using Clifford addition.
- Act is a Clifford-equivariant activation function (often a grade-wise ReLU or a sigmoid).
5. Mathematical Summary
| Feature | Scalar KAN | Clifford-KAN (CKAN) |
|---|---|---|
| Domain | $\mathbb{R}$ | $\mathcal{C}\ell_{p,q,r}$ (Multivectors) |
| Edge Op | Univariate Spline $\phi(x)$ | Multivector Spline $\Psi(u)$ |
| Interaction | Scalar Addition | Clifford Geometric Product |
Reference: Wolff, M., Alesiani, F., Duhme, C., & Jiang, X. (2026). Clifford Kolmogorov-Arnold Networks. arXiv:2602.05977.
@article{wolff2026clifford,
title={Clifford Kolmogorov-Arnold Networks},
author={Wolff, M. and Alesiani, F. and Duhme, C. and Jiang, X.},
journal={arXiv preprint arXiv:2602.05977},
year={2026},
url={https://arxiv.org/abs/2602.05977},
archivePrefix={arXiv},
eprint={2602.05977},
primaryClass={cs.LG}
}
