Horo-functions associated to atom sequences on the Wasserstein space
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Archiv der Mathematik
Horo-functions associated to atom sequences on the Wasserstein space Guomin Zhu, Hongguang Wu, and Xiaojun Cui
Abstract. On the Wasserstein space over a complete, separable, noncompact, locally compact length space, we consider the horo-functions associated to sequences of atomic measures. We show the existence of co-rays for any prescribed initial probability measure with respect to a sequence of atomic measures and show that co-rays are negative gradient curves in some sense. Some other fundamental results of this kind of functions are also obtained. Mathematics Subject Classification. 58E10, 60B10, 60H30. Keywords. Ray, Co-ray, Wasserstein space, Distance-like function, Horofunction.
1. Introduction. Recently, the Wasserstein distance Wp has presented its practicability in many topics such as geometric inference [5,7], generative adversarial networks [2], and clustering analysis [12]. Optimal transport theory [1,14,17] has significant connection to the geometry of the Wasserstein space Pp (X ), which is the set of probability measures with finite p-moments on a complete separable metric space (X , d). So the geodesic dynamics of the Wasserstein space is worthy of further study. Lisini [11] investigated absolutely continuous curves in the Wasserstein space and gave a representation of geodesic segments. For the 2-Wasserstein space over an Hadamard space, Bertrand and Kloeckner [3] studied some large-scale properties with respect to geodesic boundaries. Distance-like functions, including Busemann functions and horo-functions, play important roles in studying the global geometry of a kind of geodesic Xiaojun Cui is supported by the National Natural Science Foundation of China (Grants 11631006, 11790272, 11571166). The project is funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.
Arch. Math.
G. Zhu et al.
spaces (see e.g. [8,9,15,18]), where the notions of distance-like function and horo-function are introduced by Gromov [10]. Specifically, if the space is a noncompact complete Riemannian manifold, then the set of distance-like functions coincides with the set of viscosity solutions to the eikonal equation up to a constant [9]. So the negative gradient curves (in a weak sense) are minimal geodesics in this setting. For the generalization to more general geodesic spaces, distance-like functions are more convenient than viscosity solutions since no differential structure is required. Most properties of distance-like functions on Riemannian manifolds can be generalized without difficulty to the so-called G-space (see [6] for the original definition) which is a complete, boundedly compact, non-branching geodesic space. The Wasserstein space Pp (X ), if the ambient space X is compact, is also compact [16]; however, if X is not compact, it is not even locally compact [1, Remark 7.1.9]. This is the main difficulty of studying distance-like functions on the Wasserstein space. In the previous
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