Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo
This work addresses a bottleneck in sampling and inference methods for researchers in machine learning and statistics, presenting an incremental improvement by leveraging existing limitations.
The paper tackles the intractability of evaluating probability densities in Fokker-Planck-based particle flow methods by analyzing kernel density estimation drawbacks and transforming them into advantages, achieving applications in variational inference, kernel mean embeddings, and sequential Monte Carlo without specifying concrete numerical results.
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.