Phase retrieval in high dimensions: Statistical and computational phase transitions

We consider the phase retrieval problem of reconstructing a n-dimensional real or complex signal Xstar from m (possibly noisy) observations Yμ = | sumi=1n Phiμ i Xstari/√n|, for a large class of correlated real and complex random sensing matrices Phi, in a high-dimensional setting where m,n→∞ while α = m/n=Theta(1). First, we derive sharp asymptotics for the lowest possible estimation error achievable statistically and we unveil the existence of sharp phase transitions for the weak- and full-recovery thresholds as a function of the singular values of the matrix Phi. This is achieved by providing a rigorous proof of a result first obtained by the replica method from statistical mechanics. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at α=1 (real case) and α=2 (complex case). Secondly, we analyze the performance of the best-known polynomial time algorithm for this problem -- approximate message-passing -- establishing the existence of a statistical-to-algorithmic gap depending, again, on the spectral properties of Phi. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.