A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

Rushil Anirudh, Vinay Venkataraman, Karthikeyan Natesan Ramamurthy, Pavan Turaga

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

Abstract

Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of d-dimensional holes in the dataset, also known as the Betti-d number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a pointto-point matching, which is given by the n-Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance, for n points, the complexity grows according to O(n3). Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to O(K2), for a K K discretization of [0, 1]2. This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach.

Original languageEnglish (US)
Title of host publicationProceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016
PublisherIEEE Computer Society
Pages1023-1031
Number of pages9
ISBN (Electronic)9781467388504
DOIs
StatePublished - Dec 16 2016
Event29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016 - Las Vegas, United States
Duration: Jun 26 2016Jul 1 2016

Other

Other29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016
CountryUnited States
CityLas Vegas
Period6/26/167/1/16

Fingerprint

Statistical methods
Geometry
Probability density function
Machinery

ASJC Scopus subject areas

  • Computer Vision and Pattern Recognition
  • Electrical and Electronic Engineering

Cite this

Anirudh, R., Venkataraman, V., Ramamurthy, K. N., & Turaga, P. (2016). A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams. In Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016 (pp. 1023-1031). [7789622] IEEE Computer Society. https://doi.org/10.1109/CVPRW.2016.132

A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams. / Anirudh, Rushil; Venkataraman, Vinay; Ramamurthy, Karthikeyan Natesan; Turaga, Pavan.

Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016. IEEE Computer Society, 2016. p. 1023-1031 7789622.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Anirudh, R, Venkataraman, V, Ramamurthy, KN & Turaga, P 2016, A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams. in Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016., 7789622, IEEE Computer Society, pp. 1023-1031, 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016, Las Vegas, United States, 6/26/16. https://doi.org/10.1109/CVPRW.2016.132
Anirudh R, Venkataraman V, Ramamurthy KN, Turaga P. A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams. In Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016. IEEE Computer Society. 2016. p. 1023-1031. 7789622 https://doi.org/10.1109/CVPRW.2016.132
Anirudh, Rushil ; Venkataraman, Vinay ; Ramamurthy, Karthikeyan Natesan ; Turaga, Pavan. / A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams. Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016. IEEE Computer Society, 2016. pp. 1023-1031
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