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Rigorous Dynamics and Consistent Estimation in Arbitrarily Conditioned Linear Systems
Alyson Fletcher · Mojtaba Sahraee-Ardakan · Sundeep Rangan · Philip Schniter

Mon Dec 04 06:30 PM -- 10:30 PM (PST) @ Pacific Ballroom #183

The problem of estimating a random vector x from noisy linear measurements y=Ax+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 bad 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.

Author Information

Alyson Fletcher (UCLA)
Mojtaba Sahraee-Ardakan (UCLA)
Sundeep Rangan (NYU)
Philip Schniter (The Ohio State University)

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