Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set of spatiotemporal data, but classical methods for computing persistent homology do not respect the fact that we can not move backwards in time. In this paper,...

The persistence of homological features in simplicial complex representations of big datasets in R n resulting from Vietoris-Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r ≥ 1, in su...

Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic, the persistence landscape, was recently introduced by Bubenik. It is a functional summary, so it is easy to calculate sample means and variances, and it is st...

We are investigating the evolution of four big US stock market indexes' regular returns after 2000 technology crash and 2007-2009 financial crisis. Our approach is based on topological data processing (TDA). To identify measure phenomena occurring in multidimensional time series, we use persistence homology. obtain time-dependent point cloud sets using a sliding window, which connect space for....

In Mathematical Morphology (MM), dynamics are used to compute markers proceed for example watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) represents stability of extrema function. Since these concepts similar on functions, we studied their relationship found, proved, that they equal 1D functions. Here, pr...

(in alphabetic order by speaker surname) Speaker: Dominique Attali (CNRS, Grenoble) Title: Persistence-sensitive simplification of functions on surfaces in linear time. Abstract: Let f be a real-valued function defined on a triangulated surface S. The persistence diagram of f encodes the homological variations in the sequence of sublevel sets St = f−1(−∞, t]. A point (x, y) in the persistence d...

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and co...

The persistent homology provides a mathematical tool to describe “features” in a principled manner. The persistence algorithm proposed by Edelsbrunner et al. [9] can compute not only the persistent homology for a filtered simplicial complex, but also representative generating cycles for persistent homology groups. However, if there are dynamic changes either in the filtration or in the underlyi...

This is a survey paper of our recent results [6]. We present a novel method to detect robust and common topological structures of two geometric objects. The idea is to extend the notion of persistent homology [5, 12] to representations on a commutative triple ladder quiver. Our contributions of this paper are given as follows: (i) We prove that the commutative triple ladder quiver is representa...

The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes ...