  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.
