论文标题
理论上和计算方便的几何形状在全级相关矩阵上
Theoretically and computationally convenient geometries on full-rank correlation matrices
论文作者
论文摘要
与SPD矩阵相反，几乎没有工具可以在全级相关矩阵的开放椭圆机上执行Riemannian统计。商 - 植物度量标准最近是通过阳性对角线矩阵的一致性作用作为仿生不变度量的商而建造的。 SPD矩阵的空间一直被认为是Riemannian同质空间。相比之下，我们将SPD矩阵视为谎言组和仿射不变的度量标准是左右不变的度量标准。这种意外的新观点使我们能够概括商 - 携带公制的构建，并证明可以通过数值计算Riemannian的主要操作。但是，没有确保riemannian对数或fr {é} chet平均值的唯一性，这对于在椭圆上计算的情况不利。因此，我们在全级相关矩阵上定义了三个新的Riemannian指标家族，这些矩阵提供了Hadamard结构，包括两个平面。因此，riemannian对数和fr {é}的均值是独特的。我们还定义了一个nilpotent群结构，仿射对数和组平均值是唯一的。我们以封闭形式提供这四个结构的主要riemannian/群组运营。
In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and to show that the main Riemannian operations can be computed numerically. However, the uniqueness of the Riemannian logarithm or the Fr{é}chet mean are not ensured, which is bad for computing on the elliptope. Hence, we define three new families of Riemannian metrics on full-rank correlation matrices which provide Hadamard structures, including two flat. Thus the Riemannian logarithm and the Fr{é}chet mean are unique. We also define a nilpotent group structure for which the affine logarithm and the group mean are unique. We provide the main Riemannian/group operations of these four structures in closed form.
