Algebraic stability of zigzag persistence modules

Abstract

The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of R-valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. In this paper, we establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al. on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al. IntroductionPersistence Modules. Let Vec denote the category of vector spaces over some fixed field k, and let vec denote the subcategory of finite dimensional vector spaces. We define a persistence module to be a functor M : P → Vec, for P a poset. We will often refer to such M as a P-indexed module. If M takes values in vec, we say M is pointwise finite dimensional (p.f.d.). The P-indexed persistence modules form a category Vec P whose morphisms are the natural transformations.Persistence modules are the basic algebraic objects of study in the theory of persistent homology. The theory begins with the study of 1-D persistence modules, i.e. functors R → Vec or Z → Vec, where R and Z are taken to have the usual total orders. The structure theorem for 1-D persistence modules [25,43] tells us that the isomorphism type of a p.f.d. 1-D persistence module M is completely described by a collection B(M ) of intervals in R, called the barcode of M ; B(M ) specifies the decomposition of M into indecomposable summands. Persistent Homology. In topological data analysis, one often studies a data set by associating to the data a persistence module. To do so, we first associate to our data a filtration, i.e., a functor F : R → Top such that the map F a → F b is an inclusion whenever a ≤ b. For example, if our data is an R-valued function γ : T → R, for T a topological space, we may take F to be the sublevel set filtration S ↑ (γ), defined by, this indeed gives a filtration. If our data set is instead a point cloud, we often consider a Vietoris-Rips orČech filtration; see e.g. [10] for details. * Letting H i : Top → Vec denote the i th singular homology functor with coefficients in k, we obtain a (typically p.f.d.) persistence module H i F for any i ≥ 0. The barcodes B(H i F) serve as concise descriptors of the coarse-scale, global, non-linear geometric structure of the data set. These descriptors have been applied to many problems in science and engineering, e.g., to natural scene statistics, evolutionary biology, periodicity detection in gene expression data, sensor networks, and cluster...

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

“…Given any DMS γ X = (X , d X (·)), the simplicial filtration R lev (γ X ) : Int×R + → Simp defined as in Figure 3 is called the (spatiotemporal) Rips filtration of γ X . Definition 2.21 is based on a blend of ideas related to the Rips filtration [24,22,30] and the interlevel set persistence/categorified Reeb graphs [4,9,15,26]. The super-index "lev" in R lev (γ X ) is meant to emphasize the connection to "interlevelset persistence".…”

“…In this section we review the interleaving distance for R d -indexed functors [9,21,52]. In particular, the interleaving distance between integer-valued functions (Section 3.2) will be utilized for obtaining a computationally tractable lower bound for d dyn .…”

“…This stability implies that d Vec I between 3-dimensional persistence modules serves as a lower bound for d dyn . Since computing d Vec I between 3-dimensional persistence modules is prohibitive [9], we make use of the rank invariants/Betti-0 functions of DMSs (Definitions 2.23 and 2.24) and the interleaving distance d I between integervalued functions (Section 3.2) to obtain a lower bound for d dyn as below.…”

“…In this work, we achieve both items (a) and (b) above, described as follows.With regard to (a), we first extract invariants from a given DMS, where these invariants are in the form of 3-dimensional persistence modules of sets or vector spaces. These are obtained from a blend of ideas related to the Rips filtration [24,22,30], the single linkage hierarchical clustering (SLHC) method [16], and the interlevel set persistence/categorified Reeb graphs [4,9,15,26].We are able to prove the stability of these invariants (Theorems 4.1 and 6.17) by adapting ideas from [16,22,23]. We specifically emphasize that our stability results are a generalization of the well known stability theorems for the SLHC method [16] and the Rips filtration of a metric space [22,23]: Indeed, we show that by restricting ourselves to the class of constant DMSs, our results reduce to the standard stability theorems for static metric spaces in [16,22,23].Next, in regard to item (b) above, we address the issue of computability of the metric between invariants of DMSs.…”

“…In [7,8], Bjerkevik and Botnan show that computing the interleaving distance d I [52] between multidimensional persistence modules can in general be NP-hard. Also, since we are not guaranteed to have interval decomposability [9,17] of the 3-dimensional modules considered in this paper, we are not in a position to utilize the bottleneck distance and relevant algorithms developed by Dey and Xin [28] instead of d I .This motivates us to further simplify our invariant M X associated to a DMS (X , d X (·)), which is in the form of 3-dimensional persistence module. We focus on both the dimension function and the rank function.…”