Exponentially Fast Concentration of Vector Approximate Message Passing to its State Evolution

Vector Approximate Message Passing is a computationally-efficient iterative algorithm for estimation in high-dimensional regression problems. Due to the presence of an `Onsager' correction term in its iterates, for a wide class of N × M design matrices, namely those that are right orthogonally-invariant, the asymptotic distribution of the algorithm's estimate of the signal at any iteration can be exactly characterized in the large system limit as M/N → δ ∈ (0,∞) via a scalar recursion referred to as state evolution. In this paper, we show that appropriate functionals of the iterates in fact concentrate around their limiting values predicted by these asymptotic distributions with rates exponentially fast in N.