On the Hilbert geometry of some matrix spaces
by Frank Nielsen

Sony Computer Science Laboratories, Inc
Tokyo, Japan
https://franknielsen.github.io/

Abstract:

In this talk, we first present the dualistic structures of information geometry on the manifold of
symmetric positive-definite matrices [1,2], and discuss some algorithmic aspects [1,2,3].
We then study the Hilbert geometry induced by some bounded convex matrix spaces with applications: the elliptope of correlations matrices [4], and the Siegel disk domain [5].



References:
[1] Nielsen, Frank, and Richard Nock. "Sided and symmetrized Bregman centroids."
IEEE transactions on Information Theory 55.6 (2009): 2882-2904.

[2] Nielsen, Frank, and Rajendra Bhatia. Matrix information geometry.
Heidelberg, Germany, Springer, 2013.

[3] Arnaudon, Marc, and Frank Nielsen. "On approximating the Riemannian 1-center."
Computational Geometry 46.1 (2013): 93-104.

[4] Nielsen, Frank, and Ke Sun. "Clustering in Hilbert's projective geometry: The case studies of the probability simplex and the elliptope of correlation matrices."
Geometric Structures of Information. Springer,  2019. 297-331.

[5] Nielsen, Frank. "The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain."
Entropy 22.9 (2020): 1019.