A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

Rushil Anirudh, Vinay Venkataraman, Karthikeyan Natesan Ramamurthy, Pavan Turaga; Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, 2016, pp. 68-76

Abstract


Topological data analysis is a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature -- the number of d-dimensional holes in the dataset, also known as the Betti-d number. The persistence of these Betti numbers using persistence diagrams (PD). A common way to compare PDs is the n-Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence before computing the distance. Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs on a Hilbert Sphere. The resulting space is much more intuitive and the distance metric is correspondence-free thereby eliminating the bottleneck. It also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics etc. We report competitive results compared with the Wasserstein metric.

Related Material


[pdf]
[bibtex]
@InProceedings{Anirudh_2016_CVPR_Workshops,
author = {Anirudh, Rushil and Venkataraman, Vinay and Natesan Ramamurthy, Karthikeyan and Turaga, Pavan},
title = {A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams},
booktitle = {Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops},
month = {June},
year = {2016}
}