Phase Retrieval By Projections

Abstract

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with 2M − 1 projections of arbitrary rank in H M . Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

Projections and phase retrieval

“…We also show that this bound is sharp when N = 2 k + 1. The results of this paper answer a number of questions raised in [4].…”

“…The author and his collaborators [1,5] previously considered the problem of reconstructing a vector from the magnitudes of its frame coefficients. In this paper we answer questions raised in the paper [4] about phase retrieval from the magnitudes of orthogonal projections onto a collection of subspaces.…”

“…In this paper we consider the problem, originally raised in [4], of reconstructing a vector x (up to a global sign) from the magnitudes ||P 1 x||, ||P 2 x||, . .…”

On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

“…Fusion phase retrieval deals with the problem of recovering x upto a phase ambiguity from the measurements of the form { P i x } m i=1 , where P i : C n /R n → W i are projection operators onto the subspaces. [46] had the initial results on this problem with regards to the conditions on the subspaces and minimum number of such subspaces required for successful recovery of x under phase ambiguity.…”

Phase retrieval for sub-Gaussian measurements

“…For this problem we would like to recover a vector x ∈ F d from a finite number of quadratic measurements {x * A j x} N j=1 where each A j is a Hermitian matrix in F d×d . This is the socalled generalized phase retrieval problem, which was first studied in [22] from a theoretical angle, but earlier in special cases such as that for orthogonal projection matrices {A j } N j=1 by others [3,11,15]. To computationally recover the signal in phase retrieval, the greatest challenge comes from the nonconvexity of the objective function when it is phrased as an optimization problem.…”