Parker, J.T.
Schniter, P.
Cevher, Volkan
Bilinear Generalized Approximate Message Passingâ€”Part I: Derivation
IEEE Transactions on Signal Processing
1053-587X
10.1109/TSP.2014.2357776
62
22
5839-5853
15
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, which enables its application to matrix completion, robust PCA, dictionary learning, and related matrix-factorization problems. Here, in Part I of a two-part paper, we derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, where central-limit theorem arguments and Taylor-series approximations apply, and under the assumption of statistically independent matrix entries with known priors. In addition, we propose 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. In Part II of the paper, we will discuss the specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and we will present the results of an extensive empirical study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem.
Approximate message passing;
belief propagation;
bilinear estimation;
matrix completion;
dictionary learning;
robust principal components analysis;
matrix factorization;
Institute of Electrical and Electronics Engineers
Piscataway
2014
http://infoscience.epfl.ch/record/203543/files/06898015.pdf;