# Spectrum Truncation Power Iteration for Agnostic Matrix Phase Retrieval

Published in IEEE Trans. on Signal Processing (To appear), 2021

Recommended citation: Liu, L., Lu, S., Zhao, T. & Wang, Z. Spectrum Truncation Power Iteration for Matrix Phase Retrieval. IEEE Trans. on Signal Processing, 2021+. http://lewis-algo.com/files/TSP_matrix.pdf

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(hM∗ , Xi, ), where the predictor matrix is denoted as X ∈ R d1×d2 , link function f(·, ) is agnostic under some mild distribution assumptions on X, and  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. Download paper here