TY - GEN

T1 - On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

AU - Thaker, Parth K.

AU - Dasarathy, Gautam

AU - Nedic, Angelia

N1 - Funding Information:
This paper is based upon work supported by the National Science Foundation under Grant No. 1934766 (OAC)
Publisher Copyright:
© 2020 IEEE.

PY - 2020/6

Y1 - 2020/6

N2 - We consider the problem of recovering a complex vector x n from m quadratic measurements \left\{ {\left\langle {{A-i}{\mathbf{x}},{\mathbf{x}}} \right\rangle } \right\}-{i = 1}m. 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 \left\{ {{A-i}} \right\}-{i = 1}m 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 globally optimal point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.

AB - We consider the problem of recovering a complex vector x n from m quadratic measurements \left\{ {\left\langle {{A-i}{\mathbf{x}},{\mathbf{x}}} \right\rangle } \right\}-{i = 1}m. 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 \left\{ {{A-i}} \right\}-{i = 1}m 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 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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U2 - 10.1109/ISIT44484.2020.9174368

DO - 10.1109/ISIT44484.2020.9174368

M3 - Conference contribution

AN - SCOPUS:85090409268

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1438

EP - 1443

BT - 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2020 IEEE International Symposium on Information Theory, ISIT 2020

Y2 - 21 July 2020 through 26 July 2020

ER -