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Fully Kolmogorov-Arnold Deep Model in Medical Image Segmentation

Xingyu Qiu, Xinghua Ma, Dong Liang, Gongning Luo, Wei Wang, Kuanquan Wang, Shuo Li

TL;DR

This paper tackles the bottleneck of scaling Kolmogorov–Arnol’d networks (KANs) to deep architectures for medical image segmentation by introducing a fully KA-based model. It presents three innovations: Shared-activation KAN (SaKAN) to simplify parameterization and increase training samples per function, Grad-Free Spline to detach spline gradients and reduce memory, and ALL U-KAN, the first representative deep model with all FC and Conv layers replaced by KA-based layers. Evaluations on BUSI, GlaS, and CVC-ClinicDB demonstrate superior segmentation performance and substantial memory and parameter efficiency relative to traditional networks and partially KA-based approaches. The work unlocks new possibilities for deep KA-based architectures in complex visual tasks, albeit with slower training due to spline computations, suggesting future work on further accelerating these components. Overall, the proposed framework achieves state-of-the-art performance in three medical segmentation tasks while dramatically reducing resource demands, enabling deeper exploration of KA-based models in practice.

Abstract

Deeply stacked KANs are practically impossible due to high training difficulties and substantial memory requirements. Consequently, existing studies can only incorporate few KAN layers, hindering the comprehensive exploration of KANs. This study overcomes these limitations and introduces the first fully KA-based deep model, demonstrating that KA-based layers can entirely replace traditional architectures in deep learning and achieve superior learning capacity. Specifically, (1) the proposed Share-activation KAN (SaKAN) reformulates Sprecher's variant of Kolmogorov-Arnold representation theorem, which achieves better optimization due to its simplified parameterization and denser training samples, to ease training difficulty, (2) this paper indicates that spline gradients contribute negligibly to training while consuming huge GPU memory, thus proposes the Grad-Free Spline to significantly reduce memory usage and computational overhead. (3) Building on these two innovations, our ALL U-KAN is the first representative implementation of fully KA-based deep model, where the proposed KA and KAonv layers completely replace FC and Conv layers. Extensive evaluations on three medical image segmentation tasks confirm the superiority of the full KA-based architecture compared to partial KA-based and traditional architectures, achieving all higher segmentation accuracy. Compared to directly deeply stacked KAN, ALL U-KAN achieves 10 times reduction in parameter count and reduces memory consumption by more than 20 times, unlocking the new explorations into deep KAN architectures.

Fully Kolmogorov-Arnold Deep Model in Medical Image Segmentation

TL;DR

This paper tackles the bottleneck of scaling Kolmogorov–Arnol’d networks (KANs) to deep architectures for medical image segmentation by introducing a fully KA-based model. It presents three innovations: Shared-activation KAN (SaKAN) to simplify parameterization and increase training samples per function, Grad-Free Spline to detach spline gradients and reduce memory, and ALL U-KAN, the first representative deep model with all FC and Conv layers replaced by KA-based layers. Evaluations on BUSI, GlaS, and CVC-ClinicDB demonstrate superior segmentation performance and substantial memory and parameter efficiency relative to traditional networks and partially KA-based approaches. The work unlocks new possibilities for deep KA-based architectures in complex visual tasks, albeit with slower training due to spline computations, suggesting future work on further accelerating these components. Overall, the proposed framework achieves state-of-the-art performance in three medical segmentation tasks while dramatically reducing resource demands, enabling deeper exploration of KA-based models in practice.

Abstract

Deeply stacked KANs are practically impossible due to high training difficulties and substantial memory requirements. Consequently, existing studies can only incorporate few KAN layers, hindering the comprehensive exploration of KANs. This study overcomes these limitations and introduces the first fully KA-based deep model, demonstrating that KA-based layers can entirely replace traditional architectures in deep learning and achieve superior learning capacity. Specifically, (1) the proposed Share-activation KAN (SaKAN) reformulates Sprecher's variant of Kolmogorov-Arnold representation theorem, which achieves better optimization due to its simplified parameterization and denser training samples, to ease training difficulty, (2) this paper indicates that spline gradients contribute negligibly to training while consuming huge GPU memory, thus proposes the Grad-Free Spline to significantly reduce memory usage and computational overhead. (3) Building on these two innovations, our ALL U-KAN is the first representative implementation of fully KA-based deep model, where the proposed KA and KAonv layers completely replace FC and Conv layers. Extensive evaluations on three medical image segmentation tasks confirm the superiority of the full KA-based architecture compared to partial KA-based and traditional architectures, achieving all higher segmentation accuracy. Compared to directly deeply stacked KAN, ALL U-KAN achieves 10 times reduction in parameter count and reduces memory consumption by more than 20 times, unlocking the new explorations into deep KAN architectures.
Paper Structure (17 sections, 1 theorem, 12 equations, 5 figures, 3 tables)

This paper contains 17 sections, 1 theorem, 12 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Deeply stacked KANs are practically impossible
  4. Existing Methods cannot exploit deep KAN
  5. Methods
  6. Shared-activation KAN eases training difficulty
  7. Variants of the Kolmogorov–Arnold Representation Theorem
  8. SaKAN reformulates the Variant Theorem
  9. Grad-Free Spline reduce memory usage
  10. ALL U-KAN
  11. Experiments
  12. Datasets
  13. Experimental Settings
  14. Qualitative and Quantitative Comparison
  15. Ablation Study
  16. ...and 2 more sections

Key Result

Theorem 1

Using Grad-Free Spline at layer $L$ has minimal impact on the optimization of current or preceding layers $L-1$. Let layer $L$ of a KAN have learnable weights $u^L$ and $v^L$, with input $\mathbf{x}^L$ and output $\mathbf{y}$, and previous layer input $\mathbf{x}^{L-1}$, expressed as $\mathbf{x}^{L-

Figures (5)

  • Figure 1: Compared to existing deep methods that are partially based on KAN (e.g., U-KAN), this paper proposes the first deep model composed fully of KA-based layers, exploring its superior performance.
  • Figure 2: KAN vs. our SaKAN
  • Figure 3: Unlike vanilla KANs, Grad-Free Spline shows that detaching spline basis gradients maintains training stability, reduces computational memory, and improves efficiency.
  • Figure 4: Overview of ALL U-KAN. The proposed KA and KAonv layers, built on SaKAN and Grad-free Splines, replace all FC and convolutional layers, yielding the first fully KA-based deep model.
  • Figure 5: Qualitative comparison. Our ALL U-KAN demonstrates superior segmentation performance overall.

Theorems & Definitions (2)

  • Theorem 1: Grad-Free Spline Preserves Optimization
  • proof