Metrics and Stabilization in One Parameter Persistence

Abstract

We propose the use of persistent homology in a supervised way. We believe homological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to a model. Fitting this model is what we believe supervised homological persistence is. We develop theory behind constructing such models and we give evidence of the usefulness of this approach in concrete data analysis tasks.

Supervised Learning Using Homology Stable Rank Kernels

“…In the case X is finite, there is a finite sequence of parameters 0 ≤ a 0 ≤ / ≤ a l such that VR ϵ (X, d) ⊂ VR τ (X, d) may fail to be the equality only if ϵ < a i ≤ τ for some i, i.e., the jumps in the Vietoris-Rips filtration can occur only when passing through some a i . Such filtrations are called tame [8,10].…”

“…In the rest of the article we explain and illustrate a framework for analyzing outcomes of persistence called hierarchical stabilization [8][9][10].…”

“…We refer to [8][9][10] for an explanation of how an action leads to a pseudometric. Here we recall how to construct a rich space of such actions.…”

“…Our aim in this article is to present how persistent homology can be combined with machine learning algorithms within a framework called hierarchical stabilization [8][9][10]. We will use hierarchical stabilization to define new persistence-based kernels and illustrate them on artificial and real-world datasets.…”

Homotopical decompositions of simplicial and Vietoris Rips complexes

“…This information is then converted into spacial information and in this article we focus on the so called Vietoris-Rips construction (Hausmann 1995) for that purpose. Homologies extracted from this space give rise to invariants of the metric space used in TDA such as persistent homology (Cagliari et al 2001;Delfinado and Edelsbrunner 1995;Edelsbrunner and Harer 2008;Ferri 1995;Frosini and Landi 1999), barcodes (Carlsson et al 2005), stable ranks (Chachólski and Riihimäki 2020;Scolamiero et al 2017), or persistent landscapes (Bubenik 2015). This conversion process, from metric into spacial information, does not in general transform the gluing of metric spaces (Taubes 1996) into homotopy push-outs and homotopy colimits of simplicial complexes.…”