Abstract
We introduce new geometrical tools to cluster data in the Siegel space. We give the expression of the Riemannian logarithm and exponential maps in the Siegel disk. These new tools help us to perform classification algorithms in the Siegel disk. We also give the expression of the sectional curvature in the Siegel disk. The sectional curvatures are negative or equal to zero, and therefore the curvature of the Siegel disk is non-positive. This result proves the convergence of the gradient descent performed when computing the mean of a set of matrix points in the Siegel disk.
This work was initiated while the author was at École Polytechnique, France. This work was further developed during a thesis with Thales LAS France and the Institute of Mathematics of Bordeaux. We thank the French MoD, DGA/AID for funding (convention CIFRE AID \(N^{\circ } 2017.0008\) & ANRT \(N^{\circ } 2017.60.0062\)).
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Cabanes, Y., Nielsen, F. (2021). Classification in the Siegel Space for Vectorial Autoregressive Data. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_74
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DOI: https://doi.org/10.1007/978-3-030-80209-7_74
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