Name
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. |
Year | Venue | DocType |
|---|---|---|
2020 | NIPS 2020 | Conference |
Volume | Citations | PageRank |
33 | 0 | 0.34 |
References | Authors | |
0 | 4 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Antoine Maillard | 1 | 2 | 2.40 |
| Bruno Loureiro | 2 | 0 | 1.69 |
| Florent Krzakala | 3 | 977 | 67.30 |
| Lenka Zdeborová | 4 | 1190 | 78.62 |