# Function Fitting Based on Kolmogorov-Arnold Theorem and Kernel Functions

arXiv: https://arxiv.org/abs/2503.23038 · HTML: https://mlsanity.com/paper/2503.23038/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 18 references that are in the map.
- First posted on arXiv 29 Mar 2025.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [On the Relationship between Self-Attention and Convolutional Layers](https://mlsanity.com/paper/1911.03584/) (2019)
- [Transformer Dissection: A Unified Understanding of Transformer's Attention via the Lens of Kernel](https://mlsanity.com/paper/1908.11775/) (2019)

## Abstract (arXiv)

This paper proposes a unified theoretical framework based on the Kolmogorov-Arnold representation theorem and kernel methods. By analyzing the mathematical relationship among kernels, B-spline basis functions in Kolmogorov-Arnold Networks (KANs) and the inner product operation in self-attention mechanisms, we establish a kernel-based feature fitting framework that unifies the two models as linear combinations of kernel functions. Under this framework, we propose a low-rank Pseudo-Multi-Head Self-Attention module (Pseudo-MHSA), which reduces the parameter count of traditional MHSA by nearly 50%. Furthermore, we design a Gaussian kernel multi-head self-attention variant (Gaussian-MHSA) to validate the effectiveness of nonlinear kernel functions in feature extraction. Experiments on the CIFAR-10 dataset demonstrate that Pseudo-MHSA model achieves performance comparable to the ViT model of the same dimensionality under the MAE framework and visualization analysis reveals their similarity of multi-head distribution patterns. Our code is publicly available.