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 least-squares estimation with a generic or “plug-in” denoiser function that can be designed 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 combined with the recently-developed Vector Approximate Message Passing (VAMP) algorithm, which is itself derived via Expectation Propagation techniques. It shown that the mean squared error of this “plug-and-play" VAMP can be exactly predicted for high-dimensional right-rotationally invariant random A and Lipschitz denoisers. The method is demonstrated on applications in image recovery and parametric bilinear estimation.