From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent

2025-12-08

AI Summary

This paper addresses the robust recovery of low-dimensional vectors—such as sparse signals or natural images—from underdetermined linear measurements corrupted by structured noise. We propose a unified generalized projected gradient descent framework that jointly leverages sparse and deep priors. To enhance training stability of the deep prior, we introduce normalized idempotent regularization; to suppress structured noise, we design a generalized back-projection strategy. Theoretically, we establish robust convergence even under model mismatch and characterize the fundamental trade-off between identifiability and stability. Experiments demonstrate that our method significantly improves noise resilience and reconstruction accuracy in both sparse signal recovery and image reconstruction tasks, while maintaining algorithmic stability and generalization capability.

Technology Category

• Machine Learning: Structured Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider the problem of recovering an unknown low-dimensional vector from noisy, underdetermined observations. We focus on the Generalized Projected Gradient Descent (GPGD) framework, which unifies traditional sparse recovery methods and modern approaches using learned deep projective priors. We extend previous convergence results to robustness to model and projection errors. We use these theoretical results to explore ways to better control stability and robustness constants. To reduce recovery errors due to measurement noise, we consider generalized back-projection strategies to adapt GPGD to structured noise, such as sparse outliers. To improve the stability of GPGD, we propose a normalized idempotent regularization for the learning of deep projective priors. We provide numerical experiments in the context of sparse recovery and image inverse problems, highlighting the trade-offs between identifiability and stability that can be achieved with such methods.

Problem

• Recovering low-dimensional vectors from noisy, underdetermined observations using GPGD.
• Enhancing stability and robustness against model errors and structured noise.
• Exploring trade-offs between identifiability and stability in sparse recovery and image inverse problems.

Innovation

• Generalized Projected Gradient Descent unifies sparse recovery and learned priors
• Normalized idempotent regularization enhances stability of deep projective priors
• Generalized back-projection strategies adapt to structured noise like outliers