A positive definite matrix is called logarithmically sparse if its matrix logarithm has many zero entries. Such matrices play a significant role in high-dimensional statistics and semidefinite optimization. In this paper, logarithmically sparse matrices are studied from the point of view of computational algebraic geometry: we present a formula for the dimension of the Zariski closure of a set of matrices with a given logarithmic sparsity pattern, give a degree bound for this variety and develop implicitization algorithms that allow to find its defining equations. We illustrate our approach with numerous examples.
翻译:如果矩阵对数的对数有许多零条目,则肯定的矩阵即称为对数稀散,这种矩阵在高维统计和半无限优化方面起着重要作用。在本文中,从计算代数几何学的角度研究对数稀散的矩阵:我们用一个公式来表示一套带有给定对数宽度模式的矩阵扎里斯基封闭的尺寸,给这种多样性定出一定的限度,并发展出可以找到其定义方程式的隐含式算法。我们用许多例子来说明我们的方法。