Related papers: Distributing Persistent Homology via Spectral Sequ…
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…
We approach the problem of the computation of persistent homology for large datasets by a divide-and-conquer strategy. Dividing the total space into separate but overlapping components, we are able to limit the total memory residency for…
We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
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…
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…
In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show…
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,…
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
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 present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
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…
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 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…
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…
Persistence diagrams are useful displays that give a summary information regarding the topological features of some phenomenon. Usually, only one persistence diagram is available, and replicated persistence diagrams are needed for…
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…
Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…