Machine-Learned Sampling of Conditioned Path Measures
This work addresses a fundamental challenge in computational statistics and machine learning for researchers and practitioners dealing with path measures, though it appears incremental as it builds on existing ideas like controlled dynamics and Wasserstein optimization.
The paper tackles the problem of sampling from posterior path measures under general prior processes by developing algorithms that combine controlled equilibrium dynamics and optimization in infinite-dimensional probability spaces. The resulting algorithms are theoretically grounded and can be integrated with neural networks for learning target trajectory ensembles without requiring data access.
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.