Related papers: A directed persistent homology theory for dissimil…
In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use…
Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being…
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…
We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov-Rozansky theory. In fact, using the Khovanov-Rozansky homology of a graph, we also obtain algebraic…
By general case we mean methods able to process simplicial sets and chain complexes not of finite type. A filtration of the object to be studied is the heart of both subjects persistent homology and spectral sequences. In this paper we…
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…
Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
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…
In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted 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.…