Characterizing the Role of a Single Coupling Layer in Affine Normalizing Flows

Draxler, Felix; Schwarz, Jonathan; Schnörr, Christoph; Köthe, Ullrich

Deep Affine Normalizing Flows are efficient and powerful models for high-dimensional density estimation and sample generation. Yet little is known about how they succeed in approximating complex distributions, given the seemingly limited expressiveness of individual affine layers. In this work, we take a first step towards theoretical understanding by analyzing the behaviour of a single affine coupling layer under maximum likelihood loss. We show that such a layer estimates and normalizes conditional moments of the data distribution, and derive a tight lower bound on the loss depending on the orthogonal transformation of the data before the affine coupling. This bound can be used to identify the optimal orthogonal transform, yielding a layer-wise training algorithm for deep affine flows. Toy examples confirm our findings and stimulate further research by highlighting the remaining gap between layer-wise and end-to-end training of deep affine flows. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this chapter (10.1007/978-3-030-71278-5_1) contains supplementary material, which is available to authorized users.