On a Parametrization of Positive Semidefinite Matrices with Zeros
On a Parametrization of Positive Semidefinite Matrices with Zeros
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关于带零点的半正定矩阵的参数化
DOI：
10.1137/100783170
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Josephine Yu
中科院分区：
文献类型：
--
作者：
M. Drton;Josephine Yu
研究了一类具有指定零点的半正定矩阵凸锥的参数化。每个这样的锥对应于一个图，其非边确定规定的零点。这个类中的每个参数化都是一个多项式映射，与图的团上支持的单纯复形相关联。映射的像是凸锥，并且当关联图是弦图且单纯复形是图的团复形时，映射只能满射到零约束半正定矩阵的锥上.我们的主要结果给出了一个半代数描述的图像的参数化弦周期。这项工作的动机是考虑的地图对应于高斯统计模型与隐藏变量的事实。
We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose nonedges determine the prescribed zeros. Each parametrization in this class is a polynomial map associated with a simplicial complex supported on cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal and the simplicial complex is the clique complex of the graph. Our main result gives a semialgebraic description of the images of the parametrizations for chordless cycles. The work is motivated by the fact that the considered maps correspond to Gaussian statistical models with hidden variables.