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

Holden Lee, Chirag Pabbaraju, Anish Prasad Sevekari, Andrej Risteski

Keywords: affine couplings, representational power, conditioning, Langevin, dynamical systems

TL;DR: Log-concave distributions can be approximated using well-conditioned affine coupling flows.

Abstract: Affine-coupling models (Dinh et al., 2014; 2016)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 affinecouplings, 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; Zhanget 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 funda-mental 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. Interms of proof techniques, we uncover and lever-age 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). In terms of informing practice, 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.