Invertible generative models for inverse problems: mitigating representation error and dataset bias
This addresses the issue of unreliable reconstructions in inverse problems like compressive sensing for researchers and practitioners in imaging, offering a more robust prior method, though it is incremental as it builds on existing generative model approaches.
The paper tackles the problem of representation error and dataset bias in generative models used as priors for inverse imaging problems, demonstrating that invertible neural networks, which have zero representation error, yield higher accuracy than sparsity priors across most undersampling ratios and better reconstructions for images with rare features, including out-of-distribution ones.
Trained generative models have shown remarkable performance as priors for inverse problems in imaging -- for example, Generative Adversarial Network priors permit recovery of test images from 5-10x fewer measurements than sparsity priors. Unfortunately, these models may be unable to represent any particular image because of architectural choices, mode collapse, and bias in the training dataset. In this paper, we demonstrate that invertible neural networks, which have zero representation error by design, can be effective natural signal priors at inverse problems such as denoising, compressive sensing, and inpainting. Given a trained generative model, we study the empirical risk formulation of the desired inverse problem under a regularization that promotes high likelihood images, either directly by penalization or algorithmically by initialization. For compressive sensing, invertible priors can yield higher accuracy than sparsity priors across almost all undersampling ratios, and due to their lack of representation error, invertible priors can yield better reconstructions than GAN priors for images that have rare features of variation within the biased training set, including out-of-distribution natural images. We additionally compare performance for compressive sensing to unlearned methods, such as the deep decoder, and we establish theoretical bounds on expected recovery error in the case of a linear invertible model.