A Geometric Analysis of Phase Retrieval

• Published in: Foundations of computational mathematics Vol. 18; no. 5; pp. 1131 - 1198
• Main Authors: Sun, Ju, Qu, Qing, Wright, John
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2018; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Curvature; Economics; Gaussian distribution; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical research; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Optimization theory; Phase retrieval; Saddle points; Nonconvex optimization; 65K05; 11D09; Trust-region method; Phase retrieval; 90C26; 49K45; Second-order geometry; Inverse problems; Mathematical imaging; Function landscape; 94A12; Ridable saddles
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-017-9365-9

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = a k ∗ x for k = 1 , … , m , is it possible to recover x ∈ C n (i.e., length- n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors a k ’s are generic (i.i.d. complex Gaussian) and numerous enough ( m ≥ C n log 3 n ), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal x , up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.