Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference (2310.16975v2)

Abstract: We present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems. Both approaches enable conditional sampling and conditional density estimation, which are core tasks in Bayesian inference$\unicode{x2013}$particularly in the simulation-based ("likelihood-free") setting. Our methods represent the target conditional distributions as transformations of a tractable reference distribution and, therefore, fall into the framework of measure transport. Although many measure transport approaches model the transformation as COT maps, obtaining the map is computationally challenging, even in moderate dimensions. To improve scalability, our numerical algorithms use neural networks to parameterize COT maps and further exploit the structure of the COT problem. Our static approach approximates the map as the gradient of a partially input-convex neural network. It uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. Our dynamic approach approximates the conditional optimal transport via the flow map of a regularized neural ODE; compared to the static approach, it is slower to train but offers more modeling choices and can lead to faster sampling. We demonstrate both algorithms numerically, comparing them with competing state-of-the-art approaches, using benchmark datasets and simulation-based Bayesian inverse problems.