Unsupervised Riemannian Clustering of Probability Density Functions
Abstract
We present an algorithm for grouping families of probability density functions (pdfs). We exploit the fact that under the square-root re-parametrization, the space of pdfs forms a Riemannian manifold, namely the unit Hilbert sphere. An immediate consequence of this re-parametrization is that different families of pdfs form different submanifolds of the unit Hilbert sphere. Therefore, the problem of clustering pdfs reduces to the problem of clustering multiple submanifolds on the unit Hilbert sphere. We solve this problem by first learning a low-dimensional representation of the pdfs using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces. Then, by assuming that the pdfs from different groups are separated, we show that the null space of a matrix built from the local representation gives the segmentation of the pdfs. We also apply of our approach to the texture segmentation problem in computer vision.
Cite
Text
Goh and Vidal. "Unsupervised Riemannian Clustering of Probability Density Functions." European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, 2008. doi:10.1007/978-3-540-87479-9_43Markdown
[Goh and Vidal. "Unsupervised Riemannian Clustering of Probability Density Functions." European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, 2008.](https://mlanthology.org/ecmlpkdd/2008/goh2008ecmlpkdd-unsupervised/) doi:10.1007/978-3-540-87479-9_43BibTeX
@inproceedings{goh2008ecmlpkdd-unsupervised,
title = {{Unsupervised Riemannian Clustering of Probability Density Functions}},
author = {Goh, Alvina and Vidal, René},
booktitle = {European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases},
year = {2008},
pages = {377-392},
doi = {10.1007/978-3-540-87479-9_43},
url = {https://mlanthology.org/ecmlpkdd/2008/goh2008ecmlpkdd-unsupervised/}
}