Phase retrieval in high dimensions: Statistical and computational phase transitions

• URL: http://arxiv.org/abs/2006.05228v2
• Date: Fri, 23 Oct 2020 15:27:51 GMT
• Title: Phase retrieval in high dimensions: Statistical and computational phase transitions
• Authors: Antoine Maillard, Bruno Loureiro, Florent Krzakala, Lenka Zdeborov\'a
• Abstract summary: We consider the problem of reconstructing a $mathbfXstar$ from $m$ (possibly noisy) observations. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at $alpha=1$ and $alpha=2$. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval.
• Score: 27.437775143419987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $m,n\to\infty$ 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.

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