Stochastic Solutions for Linear Inverse Problems using the Prior Implicit in a Denoiser

Zahra Kadkhodaie, Eero P Simoncelli

Keywords: inverse problems, image priors, unsupervised learning, semi-supervised learning, deblurring, superresolution, compressive sensing

Abstract: Deep neural networks have provided state-of-the-art solutions for problems such as image denoising, which implicitly rely on a prior probability model of natural images. Two recent lines of work – Denoising Score Matching and Plug-and-Play – propose methodologies for drawing samples from this implicit prior and using it to solve inverse problems, respectively. Here, we develop a parsimonious and robust generalization of these ideas. We rely on a classic statistical result that shows the least-squares solution for removing additive Gaussian noise can be written directly in terms of the gradient of the log of the noisy signal density. We use this to derive a stochastic coarse-to-fine gradient ascent procedure for drawing high-probability samples from the implicit prior embedded within a CNN trained to perform blind denoising. A generalization of this algorithm to constrained sampling provides a method for using the implicit prior to solve any deterministic linear inverse problem, with no additional training, thus extending the power of supervised learning for denoising to a much broader set of problems. The algorithm relies on minimal assumptions and exhibits robust convergence over a wide range of parameter choices. To demonstrate the generality of our method, we use it to obtain state-of-the-art levels of unsupervised performance for deblurring, super-resolution, and compressive sensing.

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TL;DR: We've described a framework for using the prior implicitly embedded in a denoiser to solve other linear inverse problems by sampling.

Code: https://github.com/LabForComputationalVision/universal_inverse_problem