Rigorous Dynamics and Consistent Estimation in Arbitrarily Conditioned Linear Systems

TL;DR

This paper introduces an adaptive message-passing algorithm that reliably estimates signals from noisy linear measurements with unknown parameters, even in ill-conditioned systems, with proven asymptotic optimality.

Contribution

It presents a new adaptive VAMP algorithm that works for all right-rotationally random matrices, including poorly conditioned ones, with proven convergence and optimality guarantees.

Findings

Algorithm converges to deterministic limits predicted by state evolution.

Estimates match Bayes-optimal MSE under certain conditions.

Applicable to a broad class of ill-conditioned matrices.

Abstract

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.