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

TL;DR

This paper introduces KAN-PnP, a novel plug-and-play framework using Kolmogorov-Arnold Networks as priors for single-instance inverse problems, ensuring stability, convergence, and superior performance with minimal data.

Contribution

The paper presents KAN-PnP, a new method that employs Kolmogorov-Arnold Networks as denoisers for single-shot inverse problems, with theoretical guarantees and improved results.

Findings

KAN-PnP outperforms existing methods in super-resolution tasks.

The denoiser is proven to be Lipschitz continuous, ensuring stability.

The method converges reliably under key conditions.

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.