Spectrum Truncation Power Iteration for Agnostic Matrix Phase Retrieval

Agnostic matrix phase retrieval (AMPR) is a general low-rank matrix recovery problem given a set of noisy high-dimensional data samples. To be specific, AMPR is targeting at recovering an r-rank matrix
M
^*
∈ ℝ
^d1×d2
as the parametric component from n instantiations/samples of a semi-parametric model y = f(〈
M
^*
,
X
〉, ϵ), where the predictor matrix is denoted as
X
∈ ℝ
^d1×d2
, link function f(·, ϵ) is agnostic under some mild distribution assumptions on
X
, and ϵ represents the noise. In this paper, we formulate AMPR as a rank-restricted largest eigenvalue problem by applying the second-order Stein's identity and propose a new spectrum truncation power iteration (STPower) method to obtain the desired matrix efficiently. Also, we show a favorable rank recovery result by adopting the STPower method, i.e., a near-optimal statistical convergence rate under some relatively general model assumption from a wide range of applications. Extensive simulations verify our theoretical analysis and showcase the strength of STPower compared with the other existing counterparts.