发表机构
Université de Lorraine; CNRS（洛林大学; 法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在五维Vinberg锥上提出Cholesky-Vinberg和Log-Vinberg两种均值，分别基于可解群和仿射插值，具有等变性、平坦度量及显式公式，并保持稀疏结构。
AI 中文摘要
我们研究了五维Vinberg锥$\Omega$上的两种均值，该锥被视为正定$3\times3$矩阵的稀疏锥。Cholesky-Vinberg均值由与$\Omega$相关的单传递可解群$H$得到。它是$H$-等变的，其插值曲线是两个带挠率的平坦度量联络的测地线。相应的度量之一是锥的典范Hessian度量。Log-Vinberg均值通过相关的Vinberg代数（或clan）的全局对数坐标中的仿射插值定义。由此得到的黎曼度量是完备且平坦的，其加权Fréchet均值具有显式公式。两种构造都保持定义$\Omega$的零模式。
英文摘要
We study two means on the five-dimensional Vinberg cone $Ω$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $Ω$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $Ω$.
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