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On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

4 February 2020
Parth Thaker
Gautam Dasarathy
Angelia Nedić
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Abstract

We consider the problem of recovering a complex vector x∈Cn\mathbf{x}\in \mathbb{C}^nx∈Cn from mmm quadratic measurements {⟨Aix,x⟩}i=1m\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m{⟨Ai​x,x⟩}i=1m​. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes {identifiable}, and further prove isometry properties in the case when the matrices {Ai}i=1m\{A_i\}_{i=1}^m{Ai​}i=1m​ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex {optimization} formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.

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