Towards Lockfree Persistent Homology

Morozov, Dmitriy; Nigmetov, Arnur

doi:10.1145/3350755.3400244

Cited by 16 publications

(11 citation statements)

References 8 publications

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“…In a computational context, they have also been employed for parallel and multi-scale (coarse-to fine) persistence computation on the GPU by Mendoza-Smith and Tanner (2017), and in a hybrid GPU/CPU variant of PHAT and Ripser developed by Zhang et al (2019), Zhang et al (2020). Further recent implementations based on Ripser include a reimplementation in Julia developed byČufar (2020) and a lockfree shared-memory adaptation of Ripser written by Morozov and Nigmetov (2020).…”

Section: Implicit Matrix Reductionmentioning

confidence: 99%

Ripser: efficient computation of Vietoris–Rips persistence barcodes

Bauer

2021

J Appl. and Comput. Topology

235

213

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We present an algorithm for the computation of Vietoris–Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.

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Section: Implicit Matrix Reductionmentioning

confidence: 99%

Ripser: efficient computation of Vietoris–Rips persistence barcodes

Bauer

2021

J Appl. and Comput. Topology

235

213

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“…The algorithms we used in this work for computing the homology groups did not take advantage of parallelism and available implementations are memory-intensive. Recent advances in computational homology have focused on leveraging massively parallel architectures to reduce the computation time by one to two orders of magnitude [45], [46]. We plan to evaluate and leverage these in future work.…”

Section: A Limitations and Future Workmentioning

confidence: 99%

Memory Clustering Using Persistent Homology for Multimodality- and Discontinuity-Sensitive Learning of Optimal Control Warm-Starts

et al. 2021

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“…We will focus primarily on sequential approaches to persistent homology computation. Other, non-sequential approaches include the chunk algorithm [3], spectral sequence procedures [46,22], Morse-theoretic batch reduction [32,33,58,6,29,34,48,59,21], distributed algorithms [4,53,44], GPU acceleration [63,38], streaming [41], and homotopy collapse [9,20,8]. There are closely related techniques in matrix factorization and zigzag persistence [50,11,10].…”

Section: Related Literaturementioning

confidence: 99%

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Hang¹,

Giusti²,

Ziegelmeier³

et al. 2021

Preprint

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Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -and indeed, linear -algebra. However, matrices in this domain are typically large, with rows and columns numbered in billions. Low-rank approximation of such arrays typically destroys essential information; thus, new mathematical and computational paradigms are needed for very large, sparse matrices.We present the U-match matrix factorization scheme to address this challenge. U-match has two desirable features. First, it admits a compressed storage format that reduces the number of nonzero entries held in computer memory by one or more orders of magnitude over other common factorizations. Second, it permits direct solution of diverse problems in linear and homological algebra, without decompressing matrices stored in memory. These problems include look-up and retrieval of rows and columns; evaluation of birth/death times, and extraction of generators in persistent (co)homology; and, calculation of bases for boundary and cycle subspaces of filtered chain complexes. Such bases are key to unlocking a range of other topological techniques for use in TDA, and U-match factorization is designed to make such calculations broadly accessible to practitioners.As an application, we show that individual cycle representatives in persistent homology can be retrieved at time and memory costs orders of magnitude below current state of the art, via global duality. Moreover, the algebraic machinery needed to achieve this computation already exists in many modern solvers.

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Towards Lockfree Persistent Homology

Cited by 16 publications

References 8 publications

Ripser: efficient computation of Vietoris–Rips persistence barcodes

Ripser: efficient computation of Vietoris–Rips persistence barcodes

Memory Clustering Using Persistent Homology for Multimodality- and Discontinuity-Sensitive Learning of Optimal Control Warm-Starts

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

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