A Geometric Analysis of Phase Retrieval

Understand

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1, \dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (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 theoretical explanations.
• In this paper, we take a step towards bridging this gap.
• We prove that when the measurement vectors $\mathbf a_k$'s are generic (i.i.d.