Norms of Persistent Homology introduced in topological data analysis are seen as indicators system instability, analogous to the changing predictability that is captured financial market uncertainty indexes. This paper demonstrates norms from markets significant explaining uncertainty, whilst macroeconomic only explainable by volatility. Meanwhile, volatility insignificant determination when en...

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–Hausdorf...

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverseimage persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence a...

We build a functorial pipeline for persistent homology. The input to this is filtered simplicial complex indexed by any finite metric lattice, and the output persistence diagram defined as Möbius inversion of its birth-death function. adapt Reeb graph edit distance each our categories prove that both functors in are 1-Lipschitz, making stable. Our constructions generalize classical diagram, set...

Topology is the subfield of mathematics that is concerned with the study of shape. Mathematicians have studied topological questions for the past 250 years. However, in just the past 15 years topology has been found to have many different applications to real world problems. One of these is to use a topological tool called persistence homology to understand and analyze high dimensional and comp...

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jabłoński, and Mrozek suggests an equivalent description of barcodes as functors R → Mch, where R is the poset category of real numbers and Mch is the category whose obje...

We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N . Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the alge...

The ability to perform shape retrieval based not only on full similarity, but also partial similarity is a key property for any content-based search engine. We prove that persistence diagrams can reveal a partial similarity between two shapes by showing a common subset of points. This can be explained using the MayerVietoris formulas that we develop for ordinary, relative and extended persisten...

Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in Rn induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvil...

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is inte...