Fundamental Limits of Weak Recovery with Applications to Phase Retrieval

In phase retrieval, we want to recover an unknown signal $${{\varvec{x}}}\in {{\mathbb {C}}}^d$$x∈Cd from n quadratic measurements of the form $$y_i = |\langle {{\varvec{a}}}_i,{{\varvec{x}}}\rangle |^2+w_i$$yi=|⟨ai,x⟩|2+wi, where $${{\varvec{a}}}_i\in {{\mathbb {C}}}^d$$ai∈Cd are known sensing vectors and $$w_i$$wi is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator $${\hat{{{\varvec{x}}}}}({{\varvec{y}}})$$x^(y) that is positively correlated with the signal $${{\varvec{x}}}$$x? We consider the case of Gaussian vectors $${{\varvec{a}}}_i$$ai. We prove that—in the high-dimensional limit—a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For $$n\le d-o(d)$$n≤d-o(d), no estimator can do significantly better than random and achieve a strictly positive correlation. For $$n\ge d+o(d)$$n≥d+o(d), a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theoretic arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper bound and lower bound generalize beyond phase retrieval to measurements $$y_i$$yi produced according to a generalized linear model. As a by-product of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.