  Triangular flows, also known as Kn\"{o}the-Rosenblatt measure couplings,
comprise an important building block of normalizing flow models for generative
modeling and density estimation, including popular autoregressive flow models
such as real-valued non-volume preserving transformation models (Real NVP). We
present statistical guarantees and sample complexity bounds for triangular flow
statistical models. In particular, we establish the statistical consistency and
the finite sample convergence rates of the Kullback-Leibler estimator of the
Kn\"{o}the-Rosenblatt measure coupling using tools from empirical process
theory. Our results highlight the anisotropic geometry of function classes at
play in triangular flows, shed light on optimal coordinate ordering, and lead
to statistical guarantees for Jacobian flows. We conduct numerical experiments
on synthetic data to illustrate the practical implications of our theoretical
findings.
