On efficiency and low sample complexity in phase retrieval

In this paper we show that the problem of phase retrieval can be efficiently and provably solved via an alternating minimization algorithm suitably initialized. Our initialization is based on One Bit Phase Retrieval that we introduced in [1], where we showed that O(n log(n)) Gaussian phase-less measurements ensure robust recovery of the phase. In this paper we improve the sample complexity bound to O(n) measurements for sufficiently large n, using a variant of Matrix Bernstein concentration inequality that exploits the intrinsic dimension, together with properties of one bit phase retrieval.