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Triangular flows, also known as Kn\"{o}the-Rosenblatt measure couplings,comprise an important building block of normalizing flow models for generativemodeling and density estimation, including popular autoregressive flow modelssuch as real-valued non-volume preserving transformation models (Real NVP). Wepresent statistical guarantees and sample complexity bounds for triangular flowstatistical models. In particular, we establish the statistical consistency andthe finite sample convergence rates of the Kullback-Leibler estimator of theKn\"{o}the-Rosenblatt measure coupling using tools from empirical processtheory. Our results highlight the anisotropic geometry of function classes atplay in triangular flows, shed light on optimal coordinate ordering, and leadto statistical guarantees for Jacobian flows. We conduct numerical experimentson synthetic data to illustrate the practical implications of our theoreticalfindings.

Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates