A Geometric Analysis of Phase Retrieval

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, \(y_k = \left| \varvec{a}_k^* \varvec{x} \right| \) for \(k = 1, \ldots , m\), is it possible to recover \(\varvec{x} \in \mathbb C^n\) (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 \(\varvec{a}_k\)’s are generic (i.i.d. complex Gaussian) and numerous enough (\(m \ge C n \log ^3 n\)), 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 \(\varvec{x}\), 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.