# Robust Non-negative Proximal Gradient Algorithm for Inverse Problems

arXiv: https://arxiv.org/abs/2510.23362 · HTML: https://mlsanity.com/paper/2510.23362/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 65 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 25 of 25)

- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Simple Baselines for Image Restoration](https://mlsanity.com/paper/2204.04676/) (2022)
- [Deep Generalized Unfolding Networks for Image Restoration](https://mlsanity.com/paper/2204.13348/) (2022)
- [Deep Convolutional Neural Network for Multi-modal Image Restoration and Fusion](https://mlsanity.com/paper/1910.04066/) (2019)
- [Pan-Mamba: Effective pan-sharpening with State Space Model](https://mlsanity.com/paper/2402.12192/) (2024)
- [Joint Image Filtering with Deep Convolutional Networks](https://mlsanity.com/paper/1710.04200/) (2017)
- [A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates](https://mlsanity.com/paper/1704.07807/) (2017)
- [Neural Proximal Gradient Descent for Compressive Imaging](https://mlsanity.com/paper/1806.03963/) (2018)
- [A Dataset of Multi-Illumination Images in the Wild](https://mlsanity.com/paper/1910.08131/) (2019)
- [Pansharpening via Detail Injection Based Convolutional Neural Networks](https://mlsanity.com/paper/1806.08898/) (2018)
- [Deep Unfolding with Normalizing Flow Priors for Inverse Problems](https://mlsanity.com/paper/2107.02848/) (2021)
- [Adaptive Proximal Gradient Method for Convex Optimization](https://mlsanity.com/paper/2308.02261/) (2023)
- [Content-Adaptive Non-Local Convolution for Remote Sensing Pansharpening](https://mlsanity.com/paper/2404.07543/) (2024)
- [U2Net: A General Framework with Spatial-Spectral-Integrated Double U-Net for Image Fusion](https://mlsanity.com/paper/2212.06466/) (2022)
- [Wavelet-Assisted Multi-Frequency Attention Network for Pansharpening](https://mlsanity.com/paper/2502.04903/) (2025)
- [Unsharp Mask Guided Filtering](https://mlsanity.com/paper/2106.01428/) (2021)
- [A General Adaptive Dual-level Weighting Mechanism for Remote Sensing Pansharpening](https://mlsanity.com/paper/2503.13214/) (2025)
- [Anisotropic Proximal Gradient](https://mlsanity.com/paper/2210.15531/) (2022)
- [Douglas-Rachford Splitting: Complexity Estimates and Accelerated Variants](https://mlsanity.com/paper/1407.6723/) (2014)
- [A Riemannian Proximal Newton Method](https://mlsanity.com/paper/2304.04032/) (2023)
- [Proximal Gradient Algorithms: Applications in Signal Processing](https://mlsanity.com/paper/1803.01621/) (2018)
- [On the Proximal Gradient Algorithm with Alternated Inertia](https://mlsanity.com/paper/1801.05589/) (2018)
- [A Proximal-Gradient Method for Constrained Optimization](https://mlsanity.com/paper/2404.07460/) (2024)
- [A Low-rank Projected Proximal Gradient Method for Spectral Compressed Sensing](https://mlsanity.com/paper/2405.07739/) (2024)

## Abstract (arXiv)

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.