Title: A Theoretical Perspective of Solving Phaseless Compressed Sensing via Its Nonconvex Relaxation,
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.