Performance of real phase retrieval

This paper analyzes the mean-square error performance of the popular PhaseLift algorithm for phase retrieval, which is the problem of recovering an unknown signal from the magnitudes of a collection of linear measurements of the signal. This problem arises in many physical systems where only magnitudes can be measured. Our analysis approach is based on a novel comparison lemma, which upper bounds the performance of the convex-optimization-based PhaseLift algorithm in terms of an auxiliary convex optimization algorithm which is much more amenable to analysis. An upshot of our analysis is that an n-dimensional unknown signal can be recovered from the magnitudes of cn random linear measurements, where c > 1 is a constant (which empirically appears to be 3). This improves the best known earlier results which could only guarantee signal recovery with O(n log n) magnitude-only measurements. in fact, and more explicitly, we show that the sufficient number of the measurements in the high SNR regime (which corresponds to noiseless phase retrieval) can be derived from solving a deterministic convex optimization in 3 variables.