Clustering in Hilbert simplex geometry

by Frank Nielsen and Ke Sun
An invitation to machine learning in Hilbert geometries and Birkhoff projective geometries.
Visualizing the smallest enclosing ball in the probability simplex wrt to various distances:

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 Hilbert’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.