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Vector Approximate Message Passing | IEEE Journals & Magazine | IEEE Xplore

Vector Approximate Message Passing


Abstract:

The standard linear regression (SLR) problem is to recover a vector x0 from noisy linear observations y = Ax0 + w. The approximate message passing (AMP) algorithm propose...Show More

Abstract:

The standard linear regression (SLR) problem is to recover a vector x0 from noisy linear observations y = Ax0 + w. 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.
Published in: IEEE Transactions on Information Theory ( Volume: 65, Issue: 10, October 2019)
Page(s): 6664 - 6684
Date of Publication: 13 May 2019

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