You KAN Do It in a Single Shot: Plug-and-Play Methods with Single-Instance Priors

TL;DR

KAN-PnP introduces a plug-and-play optimiser that uses Kolmogorov-Arnold Networks (KANs) as denoisers to solve inverse problems with single-instance priors. By proving the KAN denoiser is Lipschitz and showing convergence of the PnP-ADMM scheme under convex data fidelity and other mild conditions, the work enables stable single-shot learning with minimal data. Empirical results on image super-resolution and joint restoration tasks demonstrate superior accuracy and faster convergence compared to existing single-shot and pretrained priors. The approach offers a principled, data-efficient pathway for high-quality reconstructions in scenarios with limited training data, with strong potential for real-world deployment. Theoretical guarantees and ablation studies further highlight the importance of architectural choices, such as the use of B-spline bases in KANs, for stability and performance.

Abstract

The use of Plug-and-Play (PnP) methods has become a central approach for solving inverse problems, with denoisers serving as regularising priors that guide optimisation towards a clean solution. In this work, we introduce KAN-PnP, an optimisation framework that incorporates Kolmogorov-Arnold Networks (KANs) as denoisers within the Plug-and-Play (PnP) paradigm. KAN-PnP is specifically designed to solve inverse problems with single-instance priors, where only a single noisy observation is available, eliminating the need for large datasets typically required by traditional denoising methods. We show that KANs, based on the Kolmogorov-Arnold representation theorem, serve effectively as priors in such settings, providing a robust approach to denoising. We prove that the KAN denoiser is Lipschitz continuous, ensuring stability and convergence in optimisation algorithms like PnP-ADMM, even in the context of single-shot learning. Additionally, we provide theoretical guarantees for KAN-PnP, demonstrating its convergence under key conditions: the convexity of the data fidelity term, Lipschitz continuity of the denoiser, and boundedness of the regularisation functional. These conditions are crucial for stable and reliable optimisation. Our experimental results show, on super-resolution and joint optimisation, that KAN-PnP outperforms exiting methods, delivering superior performance in single-shot learning with minimal data. The method exhibits strong convergence properties, achieving high accuracy with fewer iterations.

Figures (5)

• Figure 1: The proposed KAN-PnP framework, comprising the training of a single-instance prior and its integration into an iterative optimisation algorithm.
• Figure 2: The visualisation comparison of the 'Fox' example on 2$\times$ Super-Resolution task among the UNet, FFDNet, INR and KAN denoising prior in single-shot setting.
• Figure 3: Comparing the visual result of the 'Bird' example on 4$\times$ Super-Resolution task in single-shot setting with the UNet, FFDNet, INR and KAN denoising prior.
• Figure 4: Comparison of the visual output of the Plug-and-Play framework of the 'Raccoon' example on Joint Deconvoution and Demosaicing task with traditional prior (TV), pretrained prior (UNet and FFDNet) and Single-Shor prior (Noise2Self-UNet, Noise2Self-FFDNet, INR and KAN).
• Figure 5: The visualisation comparison of the 'Baby' example on Joint Deconvoution and Demosaicing task with traditional prior (TV), pretrained prior (UNet and FFDNet) and single-shot prior (Noise2Self-UNet, Noise2Self-FFDNet, INR and KAN) in the Plug-and-Play framework.

Theorems & Definitions (3)

• theorem 1: Lipschitz Continuity of the KAN Denoiser
• proof
• theorem 2: Fixed-Point Convergence of KAN-PnP