Robust Non-negative Proximal Gradient Algorithm for Inverse Problems

TL;DR

This paper introduces a robust non-negative proximal gradient algorithm using a learnable sigmoid operator to improve stability and performance in inverse problems, especially for image reconstruction.

Contribution

It proposes a novel multiplicative update proximal gradient algorithm with convergence guarantees, replacing gradient descent with a learnable sigmoid to enforce non-negativity.

Findings

Outperforms traditional PGA and state-of-the-art methods in experiments

Enhances robustness, stability, and noise immunity in inverse problems

Successfully integrates into deep networks for multi-modal restoration

Abstract

Proximal gradient algorithms (PGA), while foundational for inverse problems like image reconstruction, often yield unstable convergence and suboptimal solutions by violating the critical non-negativity constraint. We identify the gradient descent step as the root cause of this issue, which introduces negative values and induces high sensitivity to hyperparameters. To overcome these limitations, we propose a novel multiplicative update proximal gradient algorithm (SSO-PGA) with convergence guarantees, which is designed for robustness in non-negative inverse problems. Our key innovation lies in superseding the gradient descent step with a learnable sigmoid-based operator, which inherently enforces non-negativity and boundedness by transforming traditional subtractive updates into multiplicative ones. This design, augmented by a sliding parameter for enhanced stability and convergence, not only improves robustness but also boosts expressive capacity and noise immunity. We further formulate a degradation model for multi-modal restoration and derive its SSO-PGA-based optimization algorithm, which is then unfolded into a deep network to marry the interpretability of optimization with the power of deep learning. Extensive numerical and real-world experiments demonstrate that our method significantly surpasses traditional PGA and other state-of-the-art algorithms, ensuring superior performance and stability.