论文标题
SPD（n）上Wasserstein度量的几何特征
Geometric Characteristics of Wasserstein Metric on SPD(n)
论文作者
论文摘要
Wasserstein距离，尤其是在对称的正定矩阵中，对人工智能（AI）和其他计算机科学的其他分支的发展具有广泛而深厚的影响。一个自然的想法是将$ spd \ left（n \ right）$的几何形状描述为带有Wasserstein指标的Riemannian歧管。在本文中，通过涉及纤维束，我们获得了一些局部几何量的显式表达式，包括大地测量，指数图，Riemannian Connection，Jacobi字段和曲线。此外，我们讨论了大地测量学的行为，并证明该歧管是具有非负曲率的全球测量凸，但没有结合对和切割基因座。根据算术估计，我们发现曲率可以由最小特征值控制。
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 the manifold is globally geodesic convex with non-negative curvatures but no conjugate pair and cut locus. According to arithmetic estimates, we find curvatures can be controlled by the minimal eigenvalue.
