Vector Approximate Message Passing

• Published in: IEEE transactions on information theory Vol. 65; no. 10; pp. 6664 - 6684
• Main Authors: Rangan, Sundeep, Schniter, Philip, Fletcher, Alyson K.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation algorithms; Belief propagation; compressive sensing; Covariance matrices; Evolution; inference algorithms; Iterative methods; Linear regression; Matrices (mathematics); Message passing; Minimization; random matrices; Signal processing algorithms; Singular value decomposition
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2916359

The standard linear regression (SLR) problem is to recover a vector <inline-formula> <tex-math notation="LaTeX">\mathrm {x}^{0} </tex-math></inline-formula> from noisy linear observations <inline-formula> <tex-math notation="LaTeX">\mathrm {y}=\mathrm {Ax}^{0}+\mathrm {w} </tex-math></inline-formula>. The approximate message passing (AMP) algorithm proposed by Donoho, Maleki, and Montanari is a computationally efficient iterative approach to SLR that has a remarkable property: for large i.i.d. sub-Gaussian matrices A, its per-iteration behavior is rigorously characterized by a scalar state-evolution whose fixed points, when unique, are Bayes optimal. The AMP algorithm, however, is fragile in that even small deviations from the i.i.d. sub-Gaussian model can cause the algorithm to diverge. This paper considers a "vector AMP" (VAMP) algorithm and shows that VAMP has a rigorous scalar state-evolution that holds under a much broader class of large random matrices A: those that are right-orthogonally invariant. After performing an initial singular value decomposition (SVD) of A, the per-iteration complexity of VAMP is similar to that of AMP. In addition, the fixed points of VAMP's state evolution are consistent with the replica prediction of the minimum mean-squared error derived by Tulino, Caire, Verdú, and Shamai. Numerical experiments are used to confirm the effectiveness of VAMP and its consistency with state-evolution predictions.