Phase retrieval in high dimensions: Statistical and computational phase transitions

06/09/2020

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by   Antoine Maillard, et al.

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We consider the phase retrieval problem of reconstructing a n-dimensional real or complex signal 𝐗^⋆ from m (possibly noisy) observations Y_μ = | ∑_i=1^n Φ_μ i X^⋆_i/√(n)|, for a large class of correlated real and complex random sensing matrices Φ, in a high-dimensional setting where m,n→∞ while α = m/n=Θ(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 Φ. 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 Φ. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.