An invitation to machine learning in Hilbert geometries and Birkhoff projective geometries.

- Latest paper revision

Chapter published in Geometric Structures of Information (Springer, 2019):

Clustering in Hilbertâ€™s Projective Geometry: The Case Studies of the Probability Simplex and the Elliptope of Correlation Matrices. - Reproducible research:
- Python source code (for producing the figures and experiments of the paper)
- Java source code (experimentally check the information monotonicity Hilbert's simplex metric). Formal Proof

- Related information: On Balls in a Hilbert Polygonal Geometry (SoCG MM 2017 video)

Visualizing a k-clustering:

Visualizing forward and reverse Funk balls and Hilbert balls:

Visualizing a 1-clustering (minimax, smallest enclosing ball):

Voronoi diagrams with respect to the Aitchison distance (left), the Hilbert simplex distance (middle) and its equivalent variation norm space (right)

@incollection{HSEG-2019, title={{Clustering in Hilbertâ€™s projective geometry: The case studies of the probability simplex and the elliptope of correlation matrices}}, author={Nielsen, Frank and Sun, Ke}, booktitle={Geometric Structures of Information}, pages={297--331}, year={2019}, publisher={Springer} } @article{HSG-2017, title={{Clustering in Hilbert simplex geometry}}, author={Nielsen, Frank and Sun, Ke}, journal={arXiv preprint arXiv:1704.00454}, year={2017} }

November 2018, Updated November 2021.