Nonparametric maximum likelihood approximate message passing

Abstract

Generalized approximate message passing (GAMP) is an effective algorithm for recovering signals from noisy linear measurements, assuming known a priori signal distributions. However, in practice, both the signal distribution and noise level are often unknown. The EM-GM-AMP algorithm integrates GAMP with the EM algorithm to simultaneously estimate the signal distribution and noise variance while recovering the signal. EM-GM-AMP is built on the assumption that the signal is drawn from a sparse Gaussian mixture. In this paper, we propose nonparametric maximum likelihood-AMP (NPML-AMP) for estimating an arbitrary signal distribution in this setting. In addition to providing more flexibility (and performance improvements), we argue that the nonparametric approach actually simplifies implementation and improves stability by leveraging approximate convexity, which is not available in the sparse Gaussian mixture formulation of EM-GM-AMP. We also propose a simplified noise variance estimator for use in conjunction with NPML-AMP (or EM-GM-AMP). A comprehensive numerical study validates the performance of NPML-AMP algorithm in reaching nearly minimum mean squared error (MMSE) under various signal distributions, noise levels, and undersampling ratios.