Efficient Neural Network Approaches for Conditional Optimal Transport
  with Applications in Bayesian Inference
Zheyu Oliver Wang Ricardo Baptista Youssef Marzouk Lars Ruthotto Deepanshu Verma
We present two neural network approaches that approximate the solutions of
static and dynamic conditional optimal transport (COT) problems, respectively.
Both approaches enable sampling and density estimation of conditional
probability distributions, which are core tasks in Bayesian inference. Our
methods represent the target conditional distributions as transformations of a
tractable reference distribution and, therefore, fall into the framework of
measure transport. COT maps are a canonical choice within this framework, with
desirable properties such as uniqueness and monotonicity. However, the
associated COT problems are computationally challenging, even in moderate
dimensions. To improve the scalability, our numerical algorithms leverage
neural networks to parameterize COT maps. Our methods exploit the structure of
the static and dynamic formulations of the COT problem. PCP-Map models
conditional transport maps as the gradient of a partially input convex neural
network (PICNN) and uses a novel numerical implementation to increase
computational efficiency compared to state-of-the-art alternatives. COT-Flow
models conditional transports via the flow of a regularized neural ODE; it is
slower to train but offers faster sampling. We demonstrate their effectiveness
and efficiency by comparing them with state-of-the-art approaches using
benchmark datasets and Bayesian inverse problems.