Some geometric calculations on Wasserstein space
英文摘要:
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
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John Lott
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on development of artificial intelligence (AI) and other branches of computer science. A natural idea is to describe the geometry of $SPD\left(n\right)$ as a Riemannian manifold endowed with the Wasserstein metric. In this paper, by involving the fiber bundle, we obtain explicit expressions for some locally geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures. Furthermore, we discuss the behaviour of geodesics and prove that …
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The Wasserstein--Ebin Metric: A Geometric Lift of Unbalanced Optimal Transport to the space of Riemannian metrics
本文通过动态和静态形式化方法,将不平衡最优输运从正密度空间扩展到黎曼度量空间,引入 Wasserstein--Ebin 度量,并证明体积映射定义了到 Wasserstein--Fisher--Rao 度量的黎曼淹没。
Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positive sectional curvature and is locally compact. Although it is known that -except in the case of the line- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) that …
The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures…
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Geometric calculations on probability manifolds from reciprocal relations in Master equations
本文从主方程互易关系出发,研究概率流形的几何计算,推导了Levi-Civita连接、梯度等算子,并给出了两个具体例子。
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本文提出Wasserstein空间的全局微分演算框架,证明底流形的Levi-Civita联络可提升为兼容扩展Otto度量的无挠联络,重新研究J. Lott的相关工作并得到部分不同结论。
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