Wasserstein Riemannian Geometry of Positive Definite Matrices
Wasserstein Riemannian Geometry of Positive Definite Matrices
复制标题
正定矩阵的 Wasserstein 黎曼几何
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Giovanni Pistone
中科院分区：
文献类型：
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作者：
Luigi Malagò;L. Montrucchio;Giovanni Pistone
多元非退化高斯密度上的Wasserstein距离是一个黎曼距离。在回顾了距离和度量测地线的性质之后，我们导出了正定矩阵上的黎曼度量的一个显式形式，并计算了它关于迹标积的张量形式.张量是一个矩阵，它是李雅普诺夫方程的解。计算了黎曼指数、法坐标图、黎曼梯度的显式表达式，并讨论了熵的梯度流。最后，Levi-Civita协变导数以矩阵形式与平行传输的微分方程一起计算。虽然所有的计算都是以矩阵形式给出的，但我们讨论了一种特殊的移动标架的使用。应用程序进行了简要讨论。
The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we derive an explicit form of the Riemannian metrics on positive-definite matrices and compute its tensor form with respect to the trace scalar product. The tensor is a matrix, which is the solution of a Lyapunov equation. We compute explicit form for the Riemannian exponential, the normal coordinates charts, the Riemannian gradient, and discuss the gradient flow of the entropy. Finally, the Levi-Civita covariant derivative is computed in matrix form together with the differential equation for the parallel transport. While all computations are given in matrix form, notheless we discuss the use of a special moving frame. Applications are briefly discussed.