Abstract The algebraic stability theorem for persistence modules is a central result in the theory of persistent homology. We introduce new proof technique which we use to prove n -dimensional rectangle decomposable up constant $$2n-1$$ 2 n - 1 </m...

Graphs are commonly used to encode relationships among entities, yet, their abstractness makes them incredibly difficult to analyze. Node-link diagrams are a popular method for drawing graphs. Classical techniques for the node-link diagrams include various layout methods that rely on derived information to position points, which often lack interactive exploration functionalities; and force-dire...

We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of...

Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topolog-ical features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non...

The intrinsic connection between lattice theory and topology is fairly well established. For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence ba...

Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be “topological noise,” and those with a long lifetime are considered to be “topological signal.” In this paper, we bring some statis...

Topological methods, including persistent homology, are powerful tools for analysis of high-dimensional data sets but these methods rely almost exclusively on thresholding techniques. In noisy data sets, thresholding does not always allow for the recovery of topological information. We present an easy to implement, computationally efficient pre-processing algorithm to prepare noisy point cloud ...

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with...

We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior, and to recover the eigenspaces of the endomorphism on homology induced by the self-map. The chain maps are constructed using ...

Topological data analysis (TDA) studies the shape patterns of data. Persistent homology is a widely used method in TDA that summarizes homological features at multiple scales and stores them persistence diagrams (PDs). In this paper, we propose random diagram generator (RPDG) generates sequence PDs from ones produced by RPDG underpinned model based on pairwise interacting point processes revers...