Related papers: Computing homology and persistent homology using i…
A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We…
We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of…
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…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…
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…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such…
The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input…
The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…