Phase retrieval in high dimensions: Statistical and computational phase transitions

Understand

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).