We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $mathbf{X}^{star}$ from $m$ (possibly noisy) observations $Y_mu = | sum_{i=1}^n Phi_{mu i} X^{star}_i/sqrt{n}|$, for a large class of correlated real and complex random sensing matrices $mathbf{Phi}$, in a high-dimensional setting where $m,ntoinfty$ while $alpha = m/n=Theta(1)$. 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 $mathbf{Phi}$. 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 $alpha=1$ (real case) and $alpha=2$ (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 $mathbf{Phi}$. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.
تحميل البحث