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KANO: Kolmogorov-Arnold Neural Operator for Image Super-Resolution

Chenyu Li, Danfeng Hong, Bing Zhang, Zhaojie Pan, Jocelyn Chanussot

TL;DR

The paper tackles single-image super-resolution under unknown and complex degradation by proposing Kolmogorov-Arnold Neural Operator (KANO), an interpretable SR framework. It combines a physics-inspired degradation model with Kolmogorov–Arnol'd neural networks to represent spatial and spectral characteristics via a decoupled HR content $\mathbf{X}=\mathbf{O}+\mathbf{S}$ and an explicit degradation kernel $\mathbf{K}$, optimized through an ADMM-like scheme. The approach introduces three task-specific subnetworks (K-Net, O-Net, S-Net) tailored to estimate the degradation kernel, align spectral curves, and compensate nonlinear residuals, respectively. Theoretical generalization insights for KAN and extensive experiments on natural and hyperspectral remote-sensing data demonstrate improved interpretability, robustness, and cross-domain performance relative to MLP baselines, offering a practical pathway toward physically grounded SR models.

Abstract

The highly nonlinear degradation process, complex physical interactions, and various sources of uncertainty render single-image Super-resolution (SR) a particularly challenging task. Existing interpretable SR approaches, whether based on prior learning or deep unfolding optimization frameworks, typically rely on black-box deep networks to model latent variables, which leaves the degradation process largely unknown and uncontrollable. Inspired by the Kolmogorov-Arnold theorem (KAT), we for the first time propose a novel interpretable operator, termed Kolmogorov-Arnold Neural Operator (KANO), with the application to image SR. KANO provides a transparent and structured representation of the latent degradation fitting process. Specifically, we employ an additive structure composed of a finite number of B-spline functions to approximate continuous spectral curves in a piecewise fashion. By learning and optimizing the shape parameters of these spline functions within defined intervals, our KANO accurately captures key spectral characteristics, such as local linear trends and the peak-valley structures at nonlinear inflection points, thereby endowing SR results with physical interpretability. Furthermore, through theoretical modeling and experimental evaluations across natural images, aerial photographs, and satellite remote sensing data, we systematically compare multilayer perceptrons (MLPs) and Kolmogorov-Arnold networks (KANs) in handling complex sequence fitting tasks. This comparative study elucidates the respective advantages and limitations of these models in characterizing intricate degradation mechanisms, offering valuable insights for the development of interpretable SR techniques.

KANO: Kolmogorov-Arnold Neural Operator for Image Super-Resolution

TL;DR

The paper tackles single-image super-resolution under unknown and complex degradation by proposing Kolmogorov-Arnold Neural Operator (KANO), an interpretable SR framework. It combines a physics-inspired degradation model with Kolmogorov–Arnol'd neural networks to represent spatial and spectral characteristics via a decoupled HR content and an explicit degradation kernel , optimized through an ADMM-like scheme. The approach introduces three task-specific subnetworks (K-Net, O-Net, S-Net) tailored to estimate the degradation kernel, align spectral curves, and compensate nonlinear residuals, respectively. Theoretical generalization insights for KAN and extensive experiments on natural and hyperspectral remote-sensing data demonstrate improved interpretability, robustness, and cross-domain performance relative to MLP baselines, offering a practical pathway toward physically grounded SR models.

Abstract

The highly nonlinear degradation process, complex physical interactions, and various sources of uncertainty render single-image Super-resolution (SR) a particularly challenging task. Existing interpretable SR approaches, whether based on prior learning or deep unfolding optimization frameworks, typically rely on black-box deep networks to model latent variables, which leaves the degradation process largely unknown and uncontrollable. Inspired by the Kolmogorov-Arnold theorem (KAT), we for the first time propose a novel interpretable operator, termed Kolmogorov-Arnold Neural Operator (KANO), with the application to image SR. KANO provides a transparent and structured representation of the latent degradation fitting process. Specifically, we employ an additive structure composed of a finite number of B-spline functions to approximate continuous spectral curves in a piecewise fashion. By learning and optimizing the shape parameters of these spline functions within defined intervals, our KANO accurately captures key spectral characteristics, such as local linear trends and the peak-valley structures at nonlinear inflection points, thereby endowing SR results with physical interpretability. Furthermore, through theoretical modeling and experimental evaluations across natural images, aerial photographs, and satellite remote sensing data, we systematically compare multilayer perceptrons (MLPs) and Kolmogorov-Arnold networks (KANs) in handling complex sequence fitting tasks. This comparative study elucidates the respective advantages and limitations of these models in characterizing intricate degradation mechanisms, offering valuable insights for the development of interpretable SR techniques.
Paper Structure (27 sections, 5 theorems, 24 equations, 10 figures, 5 tables)

This paper contains 27 sections, 5 theorems, 24 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Evolution of Super-Resolution Techniques
  3. Approximation of Degradation Processes: Toward Interpretable Deep Learning
  4. Challenges in Existing Interpretable Paradigm: Trade-off between Transparency and Performance
  5. Contributions of KANO
  6. Preliminaries
  7. Kolmogorov–Arnold Neural (KAN)
  8. Generalization Bound
  9. Activation Functions with Low-Rank Structure
  10. Kolmogorov-Arnold Neural Operator (KANO) in Image Super-Resolution Task
  11. Motivation
  12. Model Formulation and Optimization
  13. Network Design
  14. Initialization and Loss Function
  15. Experiments
  16. ...and 12 more sections

Key Result

Theorem 2.1

Given sample set $\{(x_i, y_i)\}_{i=1}^n$, if: then with probability at least $1 - \epsilon$, the generalization error $R(f)$ of KAN has the following upper bound: where $d_p$ represents the input dimension after flattening the blur kernel, and the constant term depends on network complexity (activation smoothness, weight norms, etc.).

Figures (10)

  • Figure 1: Illustration of the spectral fitting process using KAN. The discrete spectral measurements are transformed into an approximately continuous representation via KAN, enabling a smooth spectral profile.
  • Figure 2: Visualization of $S$, $O$, and $X$ Across Iteration Steps of KANO. (a)–(c) and (g)–(i) respectively illustrate the frequency-domain characteristics of the variables $S$, $O$, and $X$. (d)–(f) and (j)–(l) present the mean squared error (MSE) between the estimated variables $S$, $O$, and $X$ and the ground truth. In early iterations, $S$ and $O$ exhibit complementary frequency features, while in later stages, the increasing similarity between $O$ and $X$ suggests a corrective effect of $S$.
  • Figure 3: The proposed KANO network is formulated within an iterative optimization framework and comprises three task-specific subnetworks: (a) K-Net estimates the spatial degradation kernel by superimposing multiple two-dimensional spline functions, providing an interpretable representation of spatial blur. (b) O-Net integrates both 2D and 1D spline functions to align 2D spatial plane along the channel combined and piecewise approximate the spectral curves for per-pixel. (c) S-Net enforces multi-scale discrepancy compensation for the variable $\mathbf{O}$, enhancing the robustness and generalization capability under various degradation conditions.
  • Figure 4: Performance comparison of KANO and other SOTA methods on four benchmark images from Set5 (top), Set14 (upper middle), and BSD100 (bottom), respectively, with the setting of scale factor=2.
  • Figure 5: Performance comparison of KANO and other SOTA methods on the Mean Squared Error (MSE) of 3 different scenes ($\sigma$=1.5) in the CAVE dataset.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Corollary 2.10