Multiparameter persistent homology of data

Thesis

We explore two distinct topics in the field of topological data analysis: invariants and metrics for multiparameter persistence modules, and the homology of random geometric simplicial complexes.

We define a computable, stable invariant for multiparameter persistence modules, the multiparameter persistence landscape, and exemplify this invariant to be sensitive to the topology and geometry of multifiltered data sets. We prove a local bi-Lipschitz equivalence between two well-studied metrics for multiparameter persistence modules: the interleaving distance and the matching distance. A consequence of this equivalence result is that the multiparameter persistence landscape is a locally complete invariant for finitely presented multiparameter persistence modules.

Finally, we explore the asymptotic properties of Čech complexes built on compact Riemannian manifolds with non-empty boundary. We attain homological connectivity thresholds with identical leading terms. An upper threshold above which the Čech complex has homology isomorphic to the homology of the underlying manifold with high probability, and a lower threshold beneath which with high probability it does not.

Authors: Vipond, O

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: random geometric simplicial complex; persistent homology
• Subjects: Geometry, Riemannian; Algebraic topology; Mathematics