On the Universality of Coupling-based Normalizing Flows

We present a novel theoretical framework for understanding the expressive power of coupling-based normalizing flows such as RealNVP. Despite their prevalence in scientific applications, a comprehensive understanding of coupling flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. Additionally, we demonstrate that these constructions inherently lead to volume-preserving flows, a property which we show to be a fundamental constraint for expressivity. We propose a new distributional universality theorem for coupling-based normalizing flows, which overcomes several limitations of prior work. Our results support the general wisdom that the coupling architecture is expressive and provide a nuanced view for choosing the expressivity of coupling functions, bridging a gap between empirical results and theoretical understanding.