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The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

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Abstract

Persistent homology with coefficients in a field \(\mathbb {F}\) 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 with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substantially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology.

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Notes

  1. PHAT also contains pivot column representations specifically optimized for \(\mathbb {Z}_2\) coefficients using low level bit operations. These bit representations usually accelerate by a constant factor the computation of persistence on complexes involving a lot of arithmetic operations, in particular the persistent homology algorithm PHAT on S4 and Kl in Fig. 3. They do not change the asymptotic analysis done in this paragraph.

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Acknowledgments

This research is partially supported by the 7th Framework Programme for Research of the European Commission, under FET-Open Grant No. 255827 (CGL Computational Geometry Learning). This research is also partially supported by NSF (National Science Foundation, USA) Grants CCF-1048983 and CCF-1116258.

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Authors and Affiliations

  1. INRIA Sophia Antipolis-Méditerranée, Sophia Antipolis Cedex, France

    Jean-Daniel Boissonnat & Clément Maria

  2. The Ohio State University, Columbus, OH, USA

    Tamal K. Dey

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  1. Jean-Daniel Boissonnat
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  2. Tamal K. Dey
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  3. Clément Maria
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Correspondence to Clément Maria.

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Boissonnat, JD., Dey, T.K. & Maria, C. The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology. Algorithmica 73, 607–619 (2015). https://doi.org/10.1007/s00453-015-9999-4

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  • DOI: https://doi.org/10.1007/s00453-015-9999-4

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