  Flow-based models are powerful tools for designing probabilistic models with
tractable density. This paper introduces Convex Potential Flows (CP-Flow), a
natural and efficient parameterization of invertible models inspired by the
optimal transport (OT) theory. CP-Flows are the gradient map of a strongly
convex neural potential function. The convexity implies invertibility and
allows us to resort to convex optimization to solve the convex conjugate for
efficient inversion. To enable maximum likelihood training, we derive a new
gradient estimator of the log-determinant of the Jacobian, which involves
solving an inverse-Hessian vector product using the conjugate gradient method.
The gradient estimator has constant-memory cost, and can be made effectively
unbiased by reducing the error tolerance level of the convex optimization
routine. Theoretically, we prove that CP-Flows are universal density
approximators and are optimal in the OT sense. Our empirical results show that
CP-Flow performs competitively on standard benchmarks of density estimation and
variational inference.
