Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo

TL;DR

This paper explores a deterministic approach to Fokker-Planck equations, leveraging the associated velocity field for particle flow methods, and addresses practical challenges in density evaluation to enhance applications like variational inference and Monte Carlo sampling.

Contribution

It introduces a novel perspective on using the probability flow ODE for sampling and inference, overcoming intractability issues through innovative approximation strategies.

Findings

The velocity field approach enables efficient particle flow methods.

Kernel density approximation limitations can be transformed into advantages.

Applications to variational inference and sequential Monte Carlo show improved performance.

Abstract

The Fokker-Planck equation can be reformulated as a continuity equation, which naturally suggests using the associated velocity field in particle flow methods. While the resulting probability flow ODE offers appealing properties - such as defining a gradient flow of the Kullback-Leibler divergence between the current and target densities with respect to the 2-Wasserstein distance - it relies on evaluating the current probability density, which is intractable in most practical applications. By closely examining the drawbacks of approximating this density via kernel density estimation, we uncover opportunities to turn these limitations into advantages in contexts such as variational inference, kernel mean embeddings, and sequential Monte Carlo.