A generative flow for conditional sampling via optimal transport

TL;DR

This paper introduces a non-parametric generative model using optimal transport to improve conditional sampling, overcoming limitations of existing parametric models like normalizing flows and GANs.

Contribution

It proposes a novel iterative method employing block-triangular transport maps derived from optimal transport, extending previous data-driven approaches for better conditional sampling.

Findings

Successfully demonstrated on a 2D example

Effective in nonlinear ODE parameter inference

Outperforms traditional parametric models

Abstract

Sampling conditional distributions is a fundamental task for Bayesian inference and density estimation. Generative models, such as normalizing flows and generative adversarial networks, characterize conditional distributions by learning a transport map that pushes forward a simple reference (e.g., a standard Gaussian) to a target distribution. While these approaches successfully describe many non-Gaussian problems, their performance is often limited by parametric bias and the reliability of gradient-based (adversarial) optimizers to learn these transformations. This work proposes a non-parametric generative model that iteratively maps reference samples to the target. The model uses block-triangular transport maps, whose components are shown to characterize conditionals of the target distribution. These maps arise from solving an optimal transport problem with a weighted $L^2$ cost function, thereby extending the data-driven approach in [Trigila and Tabak, 2016] for conditional sampling. The proposed approach is demonstrated on a two dimensional example and on a parameter inference problem involving nonlinear ODEs.