On robust phase retrieval for sparse signals

Recovering signals from their Fourier transform magnitudes is a classical problem referred to as phase retrieval and has been around for decades. In general, the Fourier transform magnitudes do not carry enough information to uniquely identify the signal and therefore additional prior information is required. In this paper, we shall assume that the underlying signal is sparse, which is true in many applications such as X-ray crystallography, astronomical imaging, etc. Recently, several techniques involving semidefinite relaxations have been proposed for this problem, however very little analysis has been performed. The phase retrieval problem can be decomposed into two tasks - (i) identifying the support of the sparse signal from the Fourier transform magnitudes, and (ii) recovering the signal using the support information. In earlier work [13], we developed algorithms for (i) which provably recovered the support for sparsities upto O(n
¹
/
^3-ϵ
). Simulations suggest that support recovery is possible upto sparsity O(n
¹
/
^2-ϵ
). In this paper, we focus on (ii) and propose an algorithm based on semidefinite relaxation, which provably recovers the signal from its Fourier transform magnitude and support knowledge with high probability if the support size is O(n
¹
/
^2-ϵ
).