2017/06/19 by Alyson K. Fletcher, Mojtaba Sahraee-Ardakan, Fletcher, Alyson K. +5 · 1 citation
Computer Science · #Blind Source Separation Techniques #FOS: Computer and information sciences #Gaussian Processes and Bayesian Inference #Information Theory (cs.IT) #Machine Learning (cs.LG) #Target Tracking and Data Fusion in Sensor Networks
paper · pdf · doi:10.48550/arxiv.1706.06054
openalex publication_date 2017/06/19 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The problem of estimating a random vector x from noisy linear measurements y = A x + w with unknown parameters on the distributions of x and w, which must also be learned, arises in a wide range of statistical learning and linear inverse problems. We show that a computationally simple iterative message-passing algorithm can provably obtain asymptotically consistent estimates in a certain high-dimensional large-system limit (LSL) under very general parameterizations. Previous message passing techniques have required i.i.d. sub-Gaussian A matrices and often fail when the matrix is ill-conditioned. The proposed algorithm, called adaptive vector approximate message passing (Adaptive VAMP) with auto-tuning, applies to all right-rotationally random A. Importantly, this class includes matrices with arbitrarily poor conditioning. We show that the parameter estimates and mean squared error (MSE) of x in each iteration converge to deterministic limits that can be precisely predicted by a simple set of state evolution (SE) equations. In addition, a simple testable condition is provided in which the MSE matches the Bayes-optimal value predicted by the replica method. The paper thus provides a computationally simple method with provable guarantees of optimality and consistency over a large class of linear inverse problems.