Robust Non-negative Proximal Gradient Algorithm for Inverse Problems
This work addresses robustness issues in inverse problems like image reconstruction, offering a more stable solution for applications requiring non-negativity constraints, though it is incremental as it builds on existing PGA frameworks.
The paper tackled the instability and suboptimal solutions in proximal gradient algorithms for non-negative inverse problems by proposing a novel multiplicative update algorithm with a learnable sigmoid-based operator, resulting in significantly superior performance and stability compared to traditional and state-of-the-art methods.
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.