Machine-Learned Sampling of Conditioned Path Measures

TL;DR

This paper introduces algorithms that combine controlled dynamics and Wasserstein-based optimization to sample from posterior path measures, enabling neural network integration for trajectory learning without data dependence.

Contribution

It presents a novel approach that unifies controlled equilibrium dynamics with Wasserstein optimization for conditioned path sampling, applicable in high-dimensional settings.

Findings

Algorithms are theoretically grounded and effective.

Seamless integration with neural networks demonstrated.

Applicable to general prior processes and likelihoods.

Abstract

We propose algorithms for sampling from posterior path measures $P(C([0, T], \mathbb{R}^d))$ under a general prior process. This leverages ideas from (1) controlled equilibrium dynamics, which gradually transport between two path measures, and (2) optimization in $\infty$-dimensional probability space endowed with a Wasserstein metric, which can be used to evolve a density curve under the specified likelihood. The resulting algorithms are theoretically grounded and can be integrated seamlessly with neural networks for learning the target trajectory ensembles, without access to data.