Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flows

TL;DR

This paper proves that well-conditioned affine coupling normalizing flows can universally approximate all log-concave distributions, providing theoretical support for training strategies involving Gaussian padding.

Contribution

It establishes the universal approximation capability of well-conditioned affine coupling flows for log-concave distributions, connecting them with Langevin dynamics and Hénon maps.

Findings

Any log-concave distribution can be approximated by well-conditioned affine coupling flows.

Deep connections between affine coupling architectures, Langevin dynamics, and Hénon maps are uncovered.

Gaussian padding of input distributions improves conditioning and training of normalizing flows.

Abstract

Normalizing flows are a widely used class of latent-variable generative models with a tractable likelihood. Affine-coupling (Dinh et al, 2014-16) models are a particularly common type of normalizing flows, for which the Jacobian of the latent-to-observable-variable transformation is triangular, allowing the likelihood to be computed in linear time. Despite the widespread usage of affine couplings, the special structure of the architecture makes understanding their representational power challenging. The question of universal approximation was only recently resolved by three parallel papers (Huang et al.,2020;Zhang et al.,2020;Koehler et al.,2020) -- who showed reasonably regular distributions can be approximated arbitrarily well using affine couplings -- albeit with networks with a nearly-singular Jacobian. As ill-conditioned Jacobians are an obstacle for likelihood-based training, the fundamental question remains: which distributions can be approximated using well-conditioned affine coupling flows? In this paper, we show that any log-concave distribution can be approximated using well-conditioned affine-coupling flows. In terms of proof techniques, we uncover and leverage deep connections between affine coupling architectures, underdamped Langevin dynamics (a stochastic differential equation often used to sample from Gibbs measures) and H\'enon maps (a structured dynamical system that appears in the study of symplectic diffeomorphisms). Our results also inform the practice of training affine couplings: we approximate a padded version of the input distribution with iid Gaussians -- a strategy which Koehler et al.(2020) empirically observed to result in better-conditioned flows, but had hitherto no theoretical grounding. Our proof can thus be seen as providing theoretical evidence for the benefits of Gaussian padding when training normalizing flows.