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Stability of Constrained Optimization Models for Structured Signal Recovery

Yijun Zhong, Yi Shen

TL;DR

This work studies the stability of three constrained optimization models for structured signal recovery under noisy linear and nonlinear (phase retrieval) measurements. By formulating a structure-regularized feasible set through a function $f$ and a tuning parameter $\eta$, it derives non-asymptotic stability bounds that quantify robustness to $\eta$-mismatch and measurement noise, using Gaussian-width geometry of descent cones. The results establish explicit trade-offs between the number of measurements $m$ and the mismatch magnitude $|f(\bm{x}^*)-\eta|$, and cover both linear estimation and phase retrieval with practical corollaries for standard Lasso and sparse-phase retrieval. These insights offer guidance on parameter tuning and demonstrate robustness of constrained models in compressed sensing and inverse-problem applications, even when the structure parameter is not optimally chosen.

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.

Stability of Constrained Optimization Models for Structured Signal Recovery

TL;DR

This work studies the stability of three constrained optimization models for structured signal recovery under noisy linear and nonlinear (phase retrieval) measurements. By formulating a structure-regularized feasible set through a function and a tuning parameter , it derives non-asymptotic stability bounds that quantify robustness to -mismatch and measurement noise, using Gaussian-width geometry of descent cones. The results establish explicit trade-offs between the number of measurements and the mismatch magnitude , and cover both linear estimation and phase retrieval with practical corollaries for standard Lasso and sparse-phase retrieval. These insights offer guidance on parameter tuning and demonstrate robustness of constrained models in compressed sensing and inverse-problem applications, even when the structure parameter is not optimally chosen.

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.
Paper Structure (12 sections, 17 theorems, 177 equations)

This paper contains 12 sections, 17 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Road Map
  5. Notation
  6. Related Works
  7. Linear Estimation
  8. Nonlinear Estimation--Phase Retrieval
  9. Preliminaries
  10. Main Results
  11. Linear Estimation
  12. Phase Retrieval

Key Result

Lemma 3.1

Mahdi Suppose that $A\in \mathbb{R}^{m\times n}$ is a Gaussian random matrix with independent $\bm{a}_i\sim \mathcal{N}(0,I_n)$ rows, then for any subset $\mathcal{T}\subset \mathbb{R}^n$ and any $u\geq 0$, $\delta\in (0,1)$, the event holds for all $\bm{x}\in \mathcal{T}$ with probability at least $1-2\exp(-\frac{u^2}{2})$ as long as $m\geq \frac{(\omega(\mathcal{T})+u)^2}{\delta^2}.$

Theorems & Definitions (36)

  • Definition 3.1: Gaussian width
  • Definition 3.2: Descent set and Cone
  • Definition 3.3: Phase Transition Function
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 26 more