Geometric Characteristics and Stable Guarantees for Phaseless Operators and Structured Matrix Restoration
Gao Huang, Song Li
TL;DR
This work presents a unified, geometry-aware framework for stable phaseless recovery and structured matrix restoration under arbitrary noise. By introducing random embeddings through concave lifting operators $\mathcal{B}^{p}_{\Phi}$ and leveraging Talagrand’s $\gamma_{\alpha}$-functionals, the authors quantify the number of measurements required for stability and robust injectivity on arbitrary geometric sets $\mathcal{T}$ and $\mathcal{M}$. They establish high-probability stability bounds for amplitude and intensity phase retrieval and for rank-one matrix restoration, and show that empirical $\ell_q$ minimization achieves robust performance with explicit noise-scaling $\|\mathbf{z}\|_{q}/m^{1/q}$. The work also introduces adversarial-noise constructions to prove the sharpness of these bounds and demonstrates improvements over prior chaos-process bounds, providing a comprehensive, algorithm-agnostic theory with practical implications for stable phaseless recovery and structured matrix problems. The results unify phase retrieval and matrix restoration under a common probabilistic-empirical framework, revealing how geometry dictates measurement requirements and robustness.
Abstract
In this paper, we first propose a unified framework for analyzing the stability of the phaseless operators for both amplitude and intensity measurement on an arbitrary geometric set, thereby characterizing the robust performance of phase retrieval via the empirical minimization method. We introduce the random embedding of concave lifting operators to characterize the unified analysis of any geometric set. Similarly, we investigate the robust performance of structured matrix restoration problem through the robust injectivity of a linear rank one measurement operator on an arbitrary matrix set. The core of our analysis is to establish unified empirical chaos processes characterization for various matrix sets. Talagrand's $γ_α$-functionals are employed to characterize the connection between the geometric constraints and the number of measurements required for stability or robust injectivity. We also construct adversarial noise to demonstrate the sharpness of the recovery bounds derived through the empirical minimization method in the both scenarios.