论文标题
Bures-Wasserstein几何形状用于阳性遗传学矩阵及其痕量一部分
Bures-Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset
论文作者
论文摘要
在他的经典论证中，饶是使用正衡量空间和欧几里得空间之间的映射来得出与Fisher指标相对应的Riemannian距离。他在措施的整个空间和概率度量子集上获得了地狱灵魂的距离。为了强调Fisher理论与量子信息理论之间的相互作用，我们使用Riemannian淹没和商歧管将这种构建扩展到阳性遗传学矩阵的空间。 Hellinger距离的类似物证明是Bures-Wasserstein（BW）距离，这是一种以最佳传输，量子信息和优化理论出现的距离度量。首先，我们提出了与此距离相关的Riemannian度量和大地测量学的现有推导。随后，我们提出了该度量标准的riemannian距离和测量学矩阵子集的新型推导，类似于Fisher距离，以实现概率度量。
In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures-Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.
