Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
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通过最优传输和信息几何观察矩阵分解的几何
DOI：
10.3934/jgm.2017014
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
K. Modin
中科院分区：
文献类型：
--
作者：
K. Modin
概率密度的空间是一种无限的riemannian歧管，带有两种口味的里曼尼亚指标：Wasserstein和Fisher-rao。前者在最佳质量运输（OMT）中是关键的，而后者发生在信息几何形状中---统计的差异几何方法。 Riemannian结构仅限于多元高斯分布的子序列，在此诱导Riemannian指标在协方差矩阵的空间上。在这里，我们对经典矩阵分解（或因素化）进行了系统的描述，以riemannian的几何形状和兼容的主束结构。讨论了Wasserstein和Fisher-Rao几何形状。通过考虑线性转换类别和多元高斯分布类别中的OMT和信息几何形状来获得与矩阵的链接。这样，OMT与矩阵的极性分解直接相关，而信息几何形状与QR，Cholesky，频谱和奇异值分解直接相关。我们还提供了各种分解的梯度流程方程的连贯描述。大多数流在数值示例中进行了说明。该论文是先前已知和原始结果的组合。作为一项调查，它涵盖了OMT和极性分解的Riemannian几何形状（平滑和线性类别），熵梯度流量以及Fisher-Rao Metric及其在多变量高斯分布的统计歧管上的Geodesics。最初的贡献包括与各种基质分解相关的新梯度流，先前研究的同一光流的新几何解释以及基于矩阵的熵梯度流的极性分解的新证明。
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher--Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.