4,123 research outputs found

    Distributed computation of persistent homology

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    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 exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems, we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (10^9) elements within seconds on a cluster with 32 nodes using less than 10GB of memory per node

    Discrete Morse theory for computing cellular sheaf cohomology

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    Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.

    Computing multiparameter persistent homology through a discrete Morse-based approach

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    Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations

    Cellular Sheaves And Cosheaves For Distributed Topological Data Analysis

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    This dissertation proposes cellular sheaf theory as a method for decomposing data analysis problems. We present novel approaches to problems in pursuit and evasion games and topological data analysis, where cellular sheaves and cosheaves are used to extract global information from data distributed with respect to time, boolean constraints, spatial location, and density. The main contribution of this dissertation lies in the enrichment of a fundamental tool in topological data analysis, called persistent homology, through cellular sheaf theory. We present a distributed computation mechanism of persistent homology using cellular cosheaves. Our construction is an extension of the generalized Mayer-Vietoris principle to filtered spaces obtained via a sequence of spectral sequences. We discuss a general framework in which the distribution scheme can be adapted according to a user-specific property of interest. The resulting persistent homology reflects properties of the topological features, allowing the user to perform refined data analysis. Finally, we apply our construction to perform a multi-scale analysis to detect features of varying sizes that are overlooked by standard persistent homology

    Parallel decomposition of persistence modules through interval bases

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    We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field mathbbFmathbb{F}, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over mathbbRmathbb{R} by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian

    Parallel decomposition of persistence modules through interval bases

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    We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {\em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field F\mathbb{F}, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over R\mathbb{R} by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian.Comment: 37 pages, 6 figure

    Distributing Persistent Homology via Spectral Sequences

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    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 using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. Finally, we review PerMaViss, a method implementing these ideas. This procedure distributes simplicial data, while focusing on merging homological information.Comment: Comments: 31 pages, 14 figures, 1 algorithm. Changes to previous version: longer introduction added, some material has been removed due to length constraints, section 5.2 added describing the procedure of computing the persistence Mayer-Vietoris spectral sequence, followed by complexity estimates in section 5.
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