When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent metric data. Given a finite dynamic metric space (DMS), we construct the zigzag simpl...

The ability to perform not only global matching but also partial matching is investigated in computer vision and computer graphics in order to evaluate the performance of shape descriptors. In my talk I will consider the persistent homology shape descriptor, and I will illustrate some results about persistence diagrams of occluded shapes and partial shapes. The main tool is a Mayer-Vietoris for...

This paper shows a mathematical formalization, algorithms and computation software of volume optimal cycles, which are useful to understand geometric features shown in a persistence diagram. Volume optimal cycles give us concrete and optimal homologous structures, such as rings or cavities, on a given data. The key idea is the optimality on (q + 1)-chain complex for a qth homology generator. Th...

This paper presents approximate confidence intervals for each function of parameters in a Banach space based on a bootstrap algorithm. We apply kernel density approach to estimate the persistence landscape. In addition, we evaluate the quality distribution function estimator of random variables using integrated mean square error (IMSE). The results of simulation studies show a significant impro...

In this paper, we study data flows in directed networks with a hierarchical recurrent structure from a cosheaf theoretical perspective. We utilize the visual parametrization of directed recurrent programs provided in persistence diagrams for cosheaf theoretical constructions. In considering cosheaves on persistence diagrams, we link global network structure and local recurrent process data. An ...

In this work we investigate the parallel computation of homology using the Mayer-Vietoris principle. We present a two stage approach for parallelizing persistence. In the first stage, we produce a cover of the input cell complex by overlapping subspaces. In the second stage, we use this cover to build the Mayer-Vietoris blowup complex, a topological space, which organizes the various subspaces ...

In this work, we propose an alternative algorithm for annotating simplices of a simplicial complex K with sub-bases of a basis B of its p-dimensional homology group H1(K). Such annotations, summed over p-simplices in any p-cycle z, provide an expression of z in B. This allows us to answer queries of null homology and independence of p-cycles efficiently and improve the running time of the greed...

In the persistent homology of filtrations, indecomposable decompositions provide persistence diagrams. However, in almost all cases multidimensional persistence, classification modules is known to be a wild problem. One direction consider subclass interval-decomposable modules, which are direct sums interval representations. We introduce definition pre-interval representations, more natural alg...