Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916453733629952 |
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| author | Klebanov, Ilja |
| author_facet | Klebanov, Ilja |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo Klebanov, Ilja Machine Learning Numerical Analysis Statistics Theory Methodology 62D99, 65C05, 65D32, 65D40, 11K36 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. |
| title | Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo |
| topic | Machine Learning Numerical Analysis Statistics Theory Methodology 62D99, 65C05, 65D32, 65D40, 11K36 |
| url | https://arxiv.org/abs/2410.18993 |