Invertible generative models for inverse problems: mitigating\n representation error and dataset bias

Abstract

Trained generative models have shown remarkable performance as priors for\ninverse problems in imaging -- for example, Generative Adversarial Network\npriors permit recovery of test images from 5-10x fewer measurements than\nsparsity priors. Unfortunately, these models may be unable to represent any\nparticular image because of architectural choices, mode collapse, and bias in\nthe training dataset. In this paper, we demonstrate that invertible neural\nnetworks, which have zero representation error by design, can be effective\nnatural signal priors at inverse problems such as denoising, compressive\nsensing, and inpainting. Given a trained generative model, we study the\nempirical risk formulation of the desired inverse problem under a\nregularization that promotes high likelihood images, either directly by\npenalization or algorithmically by initialization. For compressive sensing,\ninvertible priors can yield higher accuracy than sparsity priors across almost\nall undersampling ratios, and due to their lack of representation error,\ninvertible priors can yield better reconstructions than GAN priors for images\nthat have rare features of variation within the biased training set, including\nout-of-distribution natural images. We additionally compare performance for\ncompressive sensing to unlearned methods, such as the deep decoder, and we\nestablish theoretical bounds on expected recovery error in the case of a linear\ninvertible model.\n