Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates
This work addresses the need for rigorous statistical foundations in normalizing flow models, offering theoretical insights that are incremental but important for improving reliability in density estimation and generative tasks.
The paper tackles the problem of providing statistical guarantees for triangular flows in generative modeling, establishing their consistency and finite sample convergence rates, with numerical experiments on synthetic data illustrating the theoretical findings.
Triangular flows, also known as Knöthe-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 Knöthe-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.