We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.‚ We introduce 'decorated' real numbers to remove ambiguity about interval endpoints.Decorations are also what make the measure theory work.‚ We define several kinds of 'tameness' for a persistence module. These occur naturally in practice. The most restrictive of these, finite type, is what is normally seen in the literature. We show how to work effectively with the less restrictive hypotheses.‚ We introduce a special notation for calculations on quiver representations. This considerably simplifies the linear algebra (for instance, in proving the 'box lemma').Our goal in introducing these ideas is to enable other authors to define persistence diagrams cleanly, in a wide variety of situations, without imposing unnecessary restrictions (such as assuming a function to be Morse). For instance, it will be seen in forthcoming work that the approach here can be used to define the levelset zigzag persistence of [5] quite broadly.This paper owes much to [7] (and its published journal version [6]), which established the existence and stability of persistence diagrams for modules whose persistence maps are of finite rank. In the present work we call these modules 'q-tame'.Traditionally, continuous persistence diagrams have been treated in one of two ways. Most commonly, one makes the aggressive assumption that the situation being studied has only finitely many 'critical values'. Alternatively, as carried out in [7] for q-tame modules, the diagram is constructed using a careful limiting process through ever-finer discretisations of the parameter. The former strategy may be appropriate when working with real-world data, where every persistence module really is finite in every way, but it excludes very common theoretical situations. The latter strategy was devised to overcome these restrictions, but unfortunately the limiting arguments turn out to be quite complicated. Our new approach gives the best of both worlds; we are able to work with broader classes of persistence modules, and we can reason about their diagrams in a clean way using arguments of a finite nature.The other debt to [7] is the recasting of stability as a statement about interleaved persistence modules, the so-called 'algebraic stability theorem'. In this paper we re-recast the result as a statement about 1-parameter families of measures. This allows us to prove stability ...
We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.
We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris-Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications. 55M25, 93A15; 55N35
In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips,Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov-Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips andČech complexes built on top of compact spaces.
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