  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.
