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

Abstract: Estimating a vector $\mathbf{x}$ from noisy linear measurements $\mathbf{Ax}+\mathbf{w}$ often requires use of prior knowledge or structural constraints on $\mathbf{x}$ 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 $\mathbf{x}$. 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 $\mathbf{A}$ and Lipschitz denoisers. The method is demonstrated on applications in image recovery and parametric bilinear estimation.