A Theoretical Perspective of Solving Phaseless Compressed Sensing via Its Nonconvex Relaxation

TL;DR

This paper provides a theoretical analysis of solving Phaseless Compressed Sensing (PCS) using nonconvex $ ext{l}_p$-minimization, establishing conditions under which solutions to the relaxed problem also solve the original.

Contribution

It introduces a theoretical framework for PCS via nonconvex $ ext{l}_p$-minimization, identifying a constant $p^*$ ensuring solution equivalence.

Findings

Existence of a constant $p^*$ in (0,1] for solution equivalence.

Derivation of $p^*$ based on data and sparsity.

Theoretical justification for using $ ext{l}_p$-minimization in PCS.

Abstract

As a natural extension of compressive sensing and the requirement of some practical problems, Phaseless Compressed Sensing (PCS) has been introduced and studied recently. Many theoretical results have been obtained for PCS with the aid of its convex relaxation. Motivated by successful applications of nonconvex relaxed methods for solving compressive sensing, in this paper, we try to investigate PCS via its nonconvex relaxation. Specifically, we relax PCS in the real context by the corresponding $\ell_p$-minimization with $p\in (0,1)$. We show that there exists a constant $p^\ast\in (0,1]$ such that for any fixed $p\in(0, p^\ast)$, every optimal solution to the $\ell_p$-minimization also solves the concerned problem; and derive an expression of such a constant $p^\ast$ by making use of the known data and the sparsity level of the concerned problem. These provide a theoretical basis for solving this class of problems via the corresponding $\ell_p$-minimization.