Wasserstein steepest descent flows of discrepancies with Riesz kernels. (English) Zbl 1540.49049
In this paper, the interest in Wasserstein flows arises from the approximation of probability measures by empirical measures when halftoning images, i.e., the gray values of an image are considered as values of a probability density function of a measure.
The Wasserstein space \({\mathcal P}_2({\mathbb R}^d)\) is defined as metric space of all Borel measures with finite second moments equipped with the Wasserstein distance. First the authors introduce Wasserstein steepest descent flows which are locally absolutely continuous curves in \({\mathcal P}_2({\mathbb R}^d)\) whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan-Kinderlehrer-Otto schemes. For a \(\lambda\)-convex functional, the Wasserstein steepest descent flow coincides with the Wasserstein gradient flow.
Further, the authors study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. They present analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. Finally, for halftoning images they provide several numerical simulations of Wasserstein steepest descent flows of discrepancies.
The Wasserstein space \({\mathcal P}_2({\mathbb R}^d)\) is defined as metric space of all Borel measures with finite second moments equipped with the Wasserstein distance. First the authors introduce Wasserstein steepest descent flows which are locally absolutely continuous curves in \({\mathcal P}_2({\mathbb R}^d)\) whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan-Kinderlehrer-Otto schemes. For a \(\lambda\)-convex functional, the Wasserstein steepest descent flow coincides with the Wasserstein gradient flow.
Further, the authors study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. They present analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. Finally, for halftoning images they provide several numerical simulations of Wasserstein steepest descent flows of discrepancies.
Reviewer: Manfred Tasche (Rostock)
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
| 35A15 | Variational methods applied to PDEs |
| 46E27 | Spaces of measures |
| 49N90 | Applications of optimal control and differential games |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 49Q22 | Optimal transportation |
Keywords:
Wasserstein steepest descent flow; Wasserstein gradient flow; Wasserstein space; Riesz kernel; interaction energy; minimizing movement scheme; Dirac measure; numerical simulations; halftoning imagesReferences:
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