Related papers: Clear and Compress: Computing Persistent Homology …
Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization…
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by…
We study the relation between the persistent homology and the spectral sequence of a filtered chain complex over a field. Our method is based on a decomposition of the persistent homology. We demonstrate that, under fairly general…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application.…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
The persistent homology pipeline includes the reduction of a, so-called, boundary matrix. We extend the work of Bauer et al. (2014) and Chen et al. (2011) where they show how to use dependencies in the boundary matrix to adapt the reduction…
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
This paper presents a new clustering algorithm for space-time data based on the concepts of topological data analysis and in particular, persistent homology. Employing persistent homology - a flexible mathematical tool from algebraic…
Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…
A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as…
We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of…
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the \v{C}ech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers.…
We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field $V$ is generated using a…