Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910561909866496 |
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| author | Hertrich, Johannes Beinert, Robert Gräf, Manuel Steidl, Gabriele |
| author_facet | Hertrich, Johannes Beinert, Robert Gräf, Manuel Steidl, Gabriele |
| contents | This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$ with the non-smooth negative distance kernel $K(x,y) = -|x-y|$, we deduce a formula for the associated functional. This functional appears to be convex, and we show that $\mathcal F_ν$ is convex along (generalized) geodesics. For the Dirac measure $ν= δ_q$, $q \in \mathbb R$ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04441 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line Hertrich, Johannes Beinert, Robert Gräf, Manuel Steidl, Gabriele Optimization and Control Numerical Analysis Probability This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$ with the non-smooth negative distance kernel $K(x,y) = -|x-y|$, we deduce a formula for the associated functional. This functional appears to be convex, and we show that $\mathcal F_ν$ is convex along (generalized) geodesics. For the Dirac measure $ν= δ_q$, $q \in \mathbb R$ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration. |
| title | Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line |
| topic | Optimization and Control Numerical Analysis Probability |
| url | https://arxiv.org/abs/2301.04441 |