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The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…
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 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…
Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that…
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…
It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we…
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…
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features…
Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex…
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…
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…
Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the…
Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many…
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations…
Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
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…
The theory of multidimensional persistence captures the topology of a multifiltration -- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a…
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 technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a…