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.

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• Bilinear Recovery using Adaptive Vector-AMP
• An Approximate Message Passing Framework for Side Information
• Asymptotic errors for convex penalized linear regression beyond Gaussian matrices
• Expectation-Maximization-Aided Hybrid Generalized Expectation Consistent for Sparse Signal Reconstruction
• Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors
• Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition
• Scalable Plug-and-Play ADMM with Convergence Guarantees
• Regularization by Denoising: Clarifications and New Interpretations
• Fully Trainable and Interpretable Non-Local Sparse Models for Image Restoration
• Compressed Coding, AMP Based Decoding and Analog Spatial Coupling
• Boosting the Performance of Plug-and-Play Priors via Denoiser Scaling