A Review of Change of Variable Formulas for Generative Modeling

Abstract

Change-of-variables (CoV) formulas allow to reduce complicated probability densities to simpler ones by a learned transformation with tractable Jacobian determinant. They are thus powerful tools for maximum-likelihood learning, Bayesian inference, outlier detection, model selection, etc. CoV formulas have been derived for a large variety of model types, but this information is scattered over many separate works. We present a systematic treatment from the unifying perspective of encoder/decoder architectures, which collects 28 CoV formulas in a single place, reveals interesting relationships between seemingly diverse methods, emphasizes important distinctions that are not always clear in the literature, and identifies surprising gaps for future research.