Plug-in Estimation in High-Dimensional Linear Inverse Problems: A Rigorous Analysis

Estimating a vector x from noisy linear measurements Ax + w often requires use of prior knowledge or structural constraints on x for accurate reconstruction. Several recent works have considered combining linear leas t-squares estimation with a generic or "plug-in" denoiser function that can be des igned in a modular manner based on the prior knowledge about x . While these methods have shown excellent performance, it has been difficult to obtain rigorous performance guarantees. This work considers plug-in denoising combine d with the recently-developed V ector Approximate Message Passing (V AMP) algor ithm, which is itself derived via Expectation Propagation techniques. It shown that the mean squared error of this "plug-and-play" V AMP can be exactly pr edicted for high-dimensional right-rotationally invariant random A and Lipschitz denoisers. The method is demonstrated on applications in image recovery an d parametric bilinear estimation.