TY - GEN

T1 - A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

AU - Anirudh, Rushil

AU - Venkataraman, Vinay

AU - Ramamurthy, Karthikeyan Natesan

AU - Turaga, Pavan

N1 - Publisher Copyright:
© 2016 IEEE.

PY - 2016/12/16

Y1 - 2016/12/16

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85010208249&partnerID=8YFLogxK

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U2 - 10.1109/CVPRW.2016.132

DO - 10.1109/CVPRW.2016.132

M3 - Conference contribution

AN - SCOPUS:85010208249

T3 - IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops

SP - 1023

EP - 1031

BT - Proceedings - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016

PB - IEEE Computer Society

T2 - 29th IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPRW 2016

Y2 - 26 June 2016 through 1 July 2016

ER -