Bilinear Generalized Approximate Message Passing—Part II: Applications

Abstract

In this paper, we extend the generalized approximate message passing (G-AMP) approach, originally proposed for high-dimensional generalized-linear regression in the context of compressive sensing, to the generalized-bilinear case. In Part I of this two-part paper, we derived our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, and proposed an adaptive damping mechanism that aids convergence under finite problem sizes, an expectation-maximization (EM)-based method to automatically tune the parameters of the assumed priors, and two rank-selection strategies. Here, in Part II, we discuss the specializations of BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and present the results of an extensive empirical study comparing BiG-AMP to state-of-the-art algorithms on each problem. Our numerical results, using both synthetic and real-world datasets, demonstrate that EM-BiG-AMP yields excellent reconstruction accuracy (often best in class) while maintaining competitive runtimes.