A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally persistence, is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, encoding topological features by so called b...

Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a statistical technique, the empirical bootstrap, to separate topological signal from topological noise. In particular, we derive confidence sets for persistence dia...

We study the multi-dimensional persistence of Carlsson and Zomorodian [3] and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicia...

Garside et al. use event history methods to analyze topological data. We provide additional background on persistent homology contrast the hazard estimators used by with traditional approaches in data analysis. In particular, former is a local method, which has advantages and disadvantages, while global. also more persistence landscapes show how complete of this statistic improves its performan...

The foundational character of certain algebraic structures as Boolean algebras and Heyting algebras is rooted in their potential to model classical and constructive logic, respectively. In this paper we discuss the contributions of algebraic logic to the study of persistence based on a new operation on the ordered structure of the input diagram of vector spaces and linear maps given by a filtra...

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated u...

In this work we test Wasserstein distance in conjunction with persistent homology, as a tool for discriminating large scale structures of simulated universes different values $\sigma_8$ cosmological parameter (present root-mean-square matter fluctuation averaged over sphere radius 8 Mpc comoving). The (a.k.a. the pair-matching distance) was proposed to measure difference between two networks te...

This thesis concerns the theoretical foundations of persistence-based topological data analysis. The primary focus of the work is on the development of theory of topological inference in the multidimensional persistence setting, where the set of available theoretical and algorithmic tools has remained comparatively underdeveloped, relative to the 1-D persistence setting. The thesis establishes ...

Abstract We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs theorems about the symmetry persistence diagrams variation such maps. In particular, we identify branching endpoints networks as sole source asymmetry relate cycle basis with a version stable marriage problem. Our analysis provides foundations fast algori...

We approach the problem of the computation of persistent homology for large datasets by a divide-and-conquer strategy. Dividing the total space into separate but overlapping components, we are able to limit the total memory residency for any part of the computation, while not degrading the overall complexity much. Locally computed persistence information is then merged from the components and t...