Kolmogorov-Arnold Convolutions: Design Principles and Empirical Studies

Kolmogorov-Arnold Convolutions: Design Principles and Empirical Studies

Introduction

The paper explores the integration of Kolmogorov-Arnold Networks (KANs) into convolutional architectures for computer vision tasks. Kolmogorov-Arnold Networks leverage the Kolmogorov-Arnold theorem, allowing the replacement of linear weight matrices with learnable splines. Unlike traditional CNNs, KANs reduce the number of parameters and computational overhead, potentially enhancing the generalization capabilities of the model. This work focuses on developing parameter-efficient designs for Kolmogorov-Arnold Convolutional layers and empirically evaluating their performance across multiple datasets and tasks such as image classification and segmentation.

Methods

Kolmogorov-Arnold Convolutions

Kolmogorov-Arnold Convolutions are formulated with univariate non-linear functions embodied by $\varphi$. A variety of functions such as splines, Radial-Basis Functions (RBFs), Wavelets, and polynomials can be used as the basis $\Tilde{\varphi}$. Replacing splines with Gram polynomials has shown significant promise, providing options for parameter-efficient fine-tuning and reducing the number of trainable parameters.

Bottleneck Kolmogorov-Arnold Convolutions

To address the high number of parameters introduced by the basis functions, a bottleneck version of the Kolmogorov-Arnold Convolutions is proposed. This design incorporates squeezing and expanding convolutions before and after applying the basis function, respectively. This allows for the effective implementation of a mixture of experts, significantly driving down the number of parameters while preserving performance.

Kolmogorov-Arnold Self-Attention and Focal Modulation

The construction of Self-KAGtention and Focal KAGention layers involves substituting traditional convolutional layers with Kolmogorov-Arnold convolutional layers. The paper suggests using bottleneck convolutions for these layers to limit memory and computational requirements.

Regularization

Regularization techniques such as weight and activation penalties and dropout placements are examined. The study also evaluates additive Gaussian noise as an alternative to dropout for regularizing these networks. It is found that noise injection, particularly in the "Full" position, provides effective regularization, boosting model robustness.

Parameter-Efficient Fine-Tuning

The proposed parameter-efficient fine-tuning (PEFT) algorithm for polynomial variants of Kolmogorov-Arnold convolutional networks focuses on refining high-order features to adapt pre-trained models to new tasks. The results indicate significant reductions in the number of trainable parameters while maintaining performance.

Empirical Studies

Image Classification

The performance of the proposed Kolmogorov-Arnold convolutional models was tested on MNIST, CIFAR10, CIFAR100, Tiny ImageNet, and ImageNet1k datasets. The empirical studies highlight:

1. Scaling: Increasing model width, via a mixture of experts, performed better than increasing depth or degree of polynomials.
2. Attention Mechanisms: Introducing Self-KAGtention improved model performance, showing the utility of these new layers.
3. Hyperparameters: Optimal hyperparameters were determined through an extensive search, leading to significant performance gains.

Segmentation

The segmentation capabilities of Kolmogorov-Arnold Convolutional models were evaluated using U-Net-like architectures on the BUSI, GlaS, and CVC-ClinicDB datasets. The redesigned models achieved state-of-the-art results, reinforcing the potential of Kolmogorov-Arnold Convolutions in segmentation tasks.

Ablation Studies

A thorough ablation study was conducted, altering various components of the Kolmogorov-Arnold convolutional layers. The findings suggest that while some modifications can degrade performance, preserving the activation residual and bottleneck structure is generally beneficial. Using linear bottlenecks instead of KAN-based bottlenecks avoided training collapse on most datasets.

Design Principles

Based on the empirical results, preliminary design principles were formulated:

1. Use Gram polynomials as the basis function.
2. Employ bottleneck versions of Kolmogorov-Arnold convolutions for scalability.
3. Scale model width over depth.
4. Adopt DenseNet-like architectures for very deep networks.
5. Integrate Self KAGNtention layers when possible.
6. Utilize U2Net architectures for segmentation tasks.
7. Regularize with $L1$/$L2$ activation and Noise Injection.

Conclusion

The research demonstrates the potential of integrating Kolmogorov-Arnold Networks into convolutional architectures, promising enhanced performance and parameter efficiency. The proposed bottleneck designs, regularization techniques, and scaling strategies offer viable paths for future model development. Traffic to further enhance performance and apply these principles in various domains will be an essential next step. This could revolutionize current practices in computer vision, driving advances in both theoretical development and practical application.