Nielsen Frank
Sony Computer Science Laboratories, Tokyo 141-0022, Japan.
Entropy (Basel). 2020 Sep 12;22(9):1019. doi: 10.3390/e22091019.
We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincaré disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincaré disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
我们研究由西格尔圆盘域诱导的希尔伯特几何，西格尔圆盘域是算子范数严格小于1的复方阵的有界开凸集。这种希尔伯特几何产生了双曲几何的克莱因圆盘模型的一种推广，此后称为西格尔 - 克莱因圆盘模型，以将其与经典的西格尔上半平面和圆盘域区分开来。在西格尔 - 克莱因圆盘中，测地线在构造上总是唯一且为欧几里得直线，这使得人们能够从计算几何设计高效的几何算法和数据结构。例如，我们展示了如何在西格尔圆盘域中逼近一组复方阵的最小包围球：我们比较了西格尔 - 庞加莱圆盘和西格尔 - 克莱因圆盘中巴多尤和克拉克森（BC）迭代核心集算法的两种推广：我们证明在西格尔 - 克莱因圆盘中进行几何计算允许人们（i）绕过西格尔 - 庞加莱圆盘模型中BC算法每次迭代所需的耗时的将中心重新定位到圆盘原点的操作，以及（ii）用从嵌套希尔伯特几何导出的有保证的上下界快速且数值逼近西格尔 - 克莱因距离。
