Plug-in Estimation in High-Dimensional Linear Inverse Problems: A Rigorous Analysis
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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 a large class of high-dimensional random A and denoisers. The method is illustrated in image reconstruction and parametric bilinear estimation.
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