Weighted persistent homology

Abstract

We introduce weighted versions of the classicalČech and Vietoris-Rips complexes. We show that a version of the Vietoris-Rips Lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.2010 Mathematics Subject Classification. 55N35 (primary), 55U99 68U10 (secondary).

A Numerical Approach for the Filtered Generalized Čech Complex

“…For example, to the generalizedČech complex C (M ) of the disk system M , the function ω : C (M ) → R, σ → µ σ which assigns theČech scale to anyČech subsystem σ ⊂ M , is a weight function, called theČech-weight function. The analogous property holds for the Vietoris-Rips complex and the Vietoris-Rips scale (see [1] for the construction of the filtered generalizedČech complex using weighted point clouds).…”

“…However, we have the following result which extends the standard Vietoris-Rips Lemma [7, Th. 2.5], to any Vietoris-Rips system (compare with [1,Th. 3.2]).…”

“…To any disk system M , the simplicial substructure C (M ) (1) given by the 1-skeleton of the generalizedČech complex of M is a basic combinatorial structure (actually, a graph) that can be easily defined, it just takes the relationship into account if every two vertices are neighbors: the set of vertices is M , and there exists an edge {D i , D j } whenever D i ∩ D j = ∅. TheČech-weight function restricted to C (M ) (1) is, in fact: ω({D i }) = 0 to every vertice, and ω({D i , D j }) = c i − c j /(r i + r j ) to any edge. In Algorithm 1 we calculate theČech-weight function ω : C M (λ) (dim) → R, for the dim-skeleton of a λ-rescaled disk system M .…”

“…2.5] using elementary geometric arguments. In [1] can be found a proof in the generalized case, following the ideas in [7].…”

DTM-Based Filtrations

“…In [3,2], a weighted version of the Vietoris-Rips complex filtration is introduced to approximate the persistence of the DTM function, and several stability and approximation results, comparable to the ones of [8], are established. Another kind of weighted Vietoris-Rips complex is presented in [1].…”