Stability of Constrained Optimization Models for Structured Signal Recovery

TL;DR

This paper analyzes the stability and robustness of three constrained optimization models for structured signal recovery, emphasizing their theoretical guarantees under noise and parameter tuning challenges.

Contribution

It provides a theoretical framework demonstrating the robustness, stability, and tradeoffs of these models, supporting their practical application in noisy and uncertain environments.

Findings

Models are robust to noise and parameter variations.

Theoretical bounds on sample complexity and mismatch error.

Tradeoff established between sample size and recovery accuracy.

Abstract

Recovering an unknown but structured signal from its measurements is a challenging problem with significant applications in fields such as imaging restoration, wireless communications, and signal processing. In this paper, we consider the inherent problem stems from the prior knowledge about the signal's structure, such as sparsity which is critical for signal recovery models. We investigate three constrained optimization models that effectively address this challenge, each leveraging distinct forms of structural priors to regularize the solution space. Our theoretical analysis demonstrates that these models exhibit robustness to noise while maintaining stability with respect to tuning parameters that is a crucial property for practical applications, when the parameter selection is often nontrivial. By providing theoretical foundations, our work supports their practical use in scenarios where measurement imperfections and model uncertainties are unavoidable. Furthermore, under mild conditions, we establish tradeoff between the sample complexity and the mismatch error.