Rigorous Dynamics and Consistent Estimation in Arbitrarily Conditioned Linear Systems

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.