On exact and robust recovery for plug-and-Play compressed sensing

Abstract

Theoretical understanding of Plug-and-Play (PnP) algorithms, where an off-the-shelf denoiser is used for image regularization, is an active research topic. In this work, we study the problems of exact and stable signal recovery from compressively sensed (CS) measurements using PnP algorithms. We focus on a class of linear denoisers for which it is possible to associate a convex regularizer Φ. We consider the CS problem of minimizing Φ(x) subject to Ax=Aξ, where A is the random sensing matrix and ξ is the ground truth. We prove that if A is Gaussian and ξ lies in the range of the associated denoiser W, then the minimizer is almost surely ξ if rank(W) is less than the number of measurements and almost never otherwise. We extend the result to subgaussian matrices, except that we can guarantee exact recovery only with high probability. For noisy measurements, we consider a robust analogue of the recovery problem and prove that the error between the recovered and the ground-truth signal is bounded by the noise strength. In particular, we derive the sample complexity of CS as a function of reconstruction error and success rate. We perform numerical experiments to validate our theoretical findings.