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 flows 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 minimum Kullback-Leibler divergence statistical 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 to illustrate the practical implications of our theoretical findings.
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OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport
A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODE's dynamics can be chosen almost arbitrarily while ensuring invertibility. Moreover, the log-determinant of the flow's Jacobian can be obtained by integrating the trace of the dynamics' Jacobian along the flow. Our proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs. First, OT-Flow leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate. Second, OT-Flow features exact trace computation with time complexity equal to trace estimators used in existing CNFs. On five high-dimensional density estimation and generative modeling tasks, OT-Flow performs competitively to state-of-the-art CNFs while on average requiring one-fourth of the number of weights with an 8x speedup in training time and 24x speedup in inference.
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- Award ID(s):
- 1751636
- PAR ID:
- 10232664
- Date Published:
- Journal Name:
- Proceedings of the AAAI Conference on Artificial Intelligence
- Volume:
- 35
- Issue:
- 1-18
- ISSN:
- 2374-3468
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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