高斯模型的对数Voronoi单元
Logarithmic Voronoi Cells for Gaussian Models
AI总结：
本文将对数Voronoi单元理论扩展至高斯统计模型，证明了其在特定模型中与对数正常态射影体重合，并研究了其分解理论及代数性质。
摘要：
我们将对数Voronoi单元理论扩展至高斯统计模型。通常，高斯模型上某一点的对数Voronoi单元是包含于其对数正常态射影体（log-normal spectrahedron）内的凸集。我们证明，对于ML度为1的模型以及线性协方差模型，这两个集合重合。特别地，对于有向图模型和无向图模型，它们均相等。我们为后一类模型引入了对数Voronoi单元的分解理论。我们还研究了协方差模型，其上的对数Voronoi单元通常严格包含于对数正常态射影体内。我们给出了二元相关模型的对数Voronoi单元的显式描述，并证明它们是半代数集。最后，我们提出猜想：无限制相关模型的对数Voronoi单元不是半代数集。
英文摘要：
We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.
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