An Expectation-Maximization Approach to Tuning Generalized Vector Approximate Message Passing

摘要

Generalized Vector Approximate Message Passing (GVAMP) is an efficient iterative algorithm for approximately minimum-mean-squared-error estimation of a random vector x ～ p_x(x) from generalized linear measurements, i.e., measurements of the form y = Q(z) where z = Ax with known A, and Q(·) is a noisy, potentially nonlinear, componentwise function. Problems of this form show up in numerous applications, including robust regression, binary classification, quantized compressive sensing, and phase retrieval. In some cases, the prior p_x and/or channel Q(·) depend on unknown deterministic parameters θ, which prevents a direct application of GVAMP. In this paper we propose a way to combine expectation maximization (EM) with GVAMP to jointly estimate x and θ. We then demonstrate how EM-GVAMP can solve the phase retrieval problem with unknown measurement-noise variance.