Vector Approximate Message Passing

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}=...

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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2916359