Maximum Likelihood on the Joint (Data, Condition) Distribution for Solving Ill-Posed Problems with Conditional Flow Models

Abstract

In the absence of conditioning information, normalizing flow models are trained using maximum likelihood. Previous extensions to conditional settings have abandoned the simple maximum-likelihood approach, as obtaining the full joint distribution of data and conditions for training purposes (or equivalently, the true conditional distribution for each point in the training dataset) is impossible. However, under special conditions that nevertheless apply to a wide class of useful problems, it is possible to obtain a simple, computationally cheap proxy with the same global minimum as the conditional likelihood. Conditional flow models trained using this proxy do not require explicit knowledge of the conditional distribution, auxiliary networks or other specific architecture, or additional loss terms beyond maximum likelihood. They provide a natural and powerful way to solve a class of ill-posed inverse problems. I demonstrate these properties on easily visualized toy problems, then use the method to successfully generate class-conditional images and to reconstruct semantically degraded images via super-resolution.