Summary of Wasserstein Gradient Flows For Moreau Envelopes Of F-divergences in Reproducing Kernel Hilbert Spaces, by Viktor Stein et al.
Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces
by Viktor Stein, Sebastian Neumayer, Nicolaj Rux, Gabriele Steidl
First submitted to arxiv on: 7 Feb 2024
Categories
- Main: Machine Learning (stat.ML)
- Secondary: Machine Learning (cs.LG); Functional Analysis (math.FA); Optimization and Control (math.OC)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary This research paper proposes a regularization technique for commonly used f-divergences in measure theory, such as the Kullback-Leibler divergence. The authors regularize the f-divergence by incorporating a squared maximum mean discrepancy (MMD) associated with a characteristic kernel K. They show that this regularization can be rewritten as the Moreau envelope of some function on the reproducing kernel Hilbert space. The authors analyze the gradients of MMD-regularized f-divergences and use their findings to study Wasserstein gradient flows of these regularized f-divergences. Numerical examples are provided for empirical measures, demonstrating the effectiveness of this approach. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper solves a problem in measure theory by finding a way to fix some limitations with commonly used f-divergences. The authors take an old idea called the Moreau envelope and use it to make these f-divergences work better. They show how this works for different types of f-divergences, including those with infinite or finite “recession constants”. The paper also compares different ways to define these f-divergences and shows that they all lead to similar results. |
Keywords
* Artificial intelligence * Regularization
