A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

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(n 3 ). 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(K 2 ), 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. arXiv:1605.08912v1 [math.AT]

Persistence codebooks for topological data analysis

“…Subsequently, a discretization of the obtained function on a fixed grid of points provides the vectorization of PDs. Anirudh et al (2016) propose an alternative approach based on the reconstruction of a certain Riemannian manifold (RM) based on PDs and its subsequent representation by a fixed-size vector. In Di Fabio and Ferri (2015), PDs are represented as the coefficients of a complex polynomial having points of PD as roots.…”

A Survey of Topological Machine Learning Methods

“…In recent years, a multitude of suitable representation methods have been introduced; we present a selection thereof, focusing on representations that have already been used in machine learning contexts. As a somewhat broad categorisation, we observe that persistence diagrams are often mapped into an auxiliary vector space , e.g., by discretisation (Anirudh et al, 2016 ; Adams et al, 2017 ), or by mapping into a (Banach- or Hilbert-) function space (Chazal et al, 2014 ; Bubenik, 2015 ; Di Fabio and Ferri, 2015 ). Alternatively, there are several kernel methods (Reininghaus et al, 2015 ; Carrière et al, 2017 ; Kusano et al, 2018 ) that enable the efficient calculation of a similarity measure between persistence diagrams.…”

Perturbation Robust Representations of Topological Persistence Diagrams

“…The proposed PTS representation is motivated from [28], where the authors create a subspace representation of blurred faces and perform face recognition on the Grassmannian. Our framework also bears some similarities to [5], where the authors use the square root representation of PDFs obtained from PDs.…”

“…These metrics are only stable under small perturbations of the data which the PDs summarize, and the complexity of computing distances between PDs grows in the order of O(n 3 ), where n is the number of points in the PD [11]. Efforts have been made to overcome these problems by attempting to map PDs to spaces that are more suitable for ML tools [5,12,52,48,51,3]. A comparison of some recent algorithms for machine learning over topological descriptors can be found in [54].…”