Abstract
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.
Highlights
German mathematician Carl Ludwig Siegel [1] (1896–1981) and Chinese mathematician Loo-KengHua [2] (1910–1985) have introduced independently the symplectic geometry in the 1940s
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
We have generalized the Klein model of hyperbolic geometry to the Siegel disk domain of complex matrices by considering the Hilbert geometry induced by the Siegel disk, an open-bounded convex complex matrix domain
Summary
German mathematician Carl Ludwig Siegel [1] (1896–1981) and Chinese mathematician Loo-Keng. To demonstrate the advantage of the Siegel–Klein disk model (Hilbert distance) over the Siegel–Poincaré disk model (Kobayashi distance), we consider approximating the Smallest Enclosing Ball (SEB) of the a set of square complex matrices in the Siegel disk domain. This problem finds potential applications in image morphology [28,38] or anomaly detection of covariance matrices [39,40]. Since the approximation factor does not depend on the dimension, this SEB approximation algorithm found many applications in machine learning [48] (e.g., in Reproducing Kernel Hilbert Spaces [49], RKHS)
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