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.

翻译：我们提出了一种新的理论框架，用于理解基于耦合的归一化流（如RealNVP）的表达能力。尽管耦合流在科学应用中广泛存在，但由于其受限的架构，对其全面理解仍较为困难。现有定理存在不足，因为它们需要使用任意病态的神经网络，这限制了实际应用性。此外，我们证明这些构造本质上会导致体积保持流，并揭示这是表达能力的根本约束。我们提出了一种新的基于耦合的归一化流分布普适性定理，克服了先前研究的若干局限性。我们的结果支持了耦合架构具有表达能力的普遍观点，并为选择耦合函数的表达能力提供了细致视角，从而弥合了实证结果与理论理解之间的差距。