Convex recovery of a structured signal from independent random linear measurements

TL;DR

This paper provides a theoretical analysis of convex programming methods for recovering structured signals from various types of random linear measurements, extending results beyond Gaussian ensembles and including phase retrieval.

Contribution

It introduces a general framework for analyzing convex recovery methods applicable to a wide range of measurement ensembles, with specific insights into phase retrieval.

Findings

Bounds on sampling complexity similar to Gaussian case

Applicable to diverse measurement ensembles

Effective analysis of phase retrieval via trace-norm minimization

Abstract

This chapter develops a theoretical analysis of the convex programming method for recovering a structured signal from independent random linear measurements. This technique delivers bounds for the sampling complexity that are similar with recent results for standard Gaussian measurements, but the argument applies to a much wider class of measurement ensembles. To demonstrate the power of this approach, the paper presents a short analysis of phase retrieval by trace-norm minimization. The key technical tool is a framework, due to Mendelson and coauthors, for bounding a nonnegative empirical process.