Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Abstract

Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS).In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrizing the collection of all DMSs.On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well known interleaving distance. In either case, the metric d can be computed in polynomial time. IntroductionStability and tractability of TDA for studying metric spaces. Finite point clouds or finite metric spaces are amongst the most common data representations considered in topological data analysis (TDA) [13,29,33]. In particular, the stability of the Single Linkage Hierarchical Clustering (SLHC) method [16] or the stability of the persistent homology of filtered Rips complexes built on metric spaces [22,23] motivates adopting these constructions when studying metric spaces arising in applications.Whereas there has been extensive applications of TDA to static metric data (thanks to the aforementioned theoretical underpinnings), there is not much study of dynamic metric data from the TDA perspective. Our motivation for considering dynamic metric data stems from the study and characterization of flocking/swarming behaviors of animals [5,36,37,39,53,57,63,69], convoys [41], moving clusters [43], or mobile groups [40,70]. In this paper, by extending ideas from [16,22,23,47,46], we aim at establishing a TDA framework for the study of dynamic metric spaces (DMSs) which comes together with stability theorems. We begin by describing and comparing relevant work with ours. Lack of an adequate metric for DMSs. In [55], Munch considers vineyards -a certain notion of time-varying persistence diagrams introduced by Cohen-Steiner et al. [25] -as signatures for dynamic point clouds. Munch, in particular, shows that vineyards are stable 1 [24] under perturbations of the input dynamic point cloud ...

Capturing dynamics of time-varying data via topology

“…We remark that this is only a pseudo-metric, not an actual metric. Indeed, as pointed out in Figure 1 of [37], two distinct time-varying metric spaces that are not qualitatively similar can have Gromov-Hausdorff distance zero from each other at each time t. The distances between multiparameter rank functions introduced in [37] are also stable with respect to more refined notions of distance between time-varying metric spaces. In this paper, we restrict attention to the weaker Definition 7.1 and show that crocker stacks, which are amenable for machine learning tasks, are furthermore a continuous topological invariant.…”

“…They prove that the resulting persistence diagram is stable under perturbations of the input dynamic graph in relation to defined distances on the dynamic graphs. In a related research direction, these authors also consider an invariant for dynamic metric spaces in [37]. They introduce a spatiotemporal filtration which can measure subtle differences between pairs of dynamic metric spaces.…”

“…By producing a 3D persistence module where one of the dimensions is not the real line but a poset, the invariant obtains higher differentiability power than a vineyard representation. Intuitively, [37] considers smoothings in both time and space. By contrast, crocker stacks focus on smoothings in space alone, with the goal of obtaining vectorizable summaries as input for machine learning tasks.…”

“…There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [19], crocker plots [55], and multiparameter rank functions [37]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack.…”

“…Thus far, we have been discussing static data. Three topological summaries of time-varying data are vineyards [19], crocker plots [55], and multiparameter rank functions [37]. As we will explain in more detail later, a vineyard is a 3D representation of persistent homology over time.…”