Geometric Characteristics and Stable Guarantees for Phaseless Operators and Structured Matrix Restoration

TL;DR

This paper develops a unified geometric framework to analyze the stability of phaseless operators and structured matrix recovery, providing sharp measurement bounds and robustness guarantees.

Contribution

It introduces a novel unified analysis method using empirical chaos processes and Talagrand's functionals for phase retrieval and matrix restoration.

Findings

Established stability bounds for phase retrieval on arbitrary sets.

Demonstrated robust injectivity conditions for structured matrix recovery.

Validated sharpness of bounds with adversarial noise constructions.

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 $\gamma_{\alpha}$-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.