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GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes

Authors Simon Zhang, Mengbai Xiao, Hao Wang

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Author Details

Simon Zhang
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Mengbai Xiao
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Hao Wang
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA

Funding

This work has been partially supported by the National Science Foundation under grants CCF-1513944, CCF-1629403, CCF-1718450, and an IBM Fellowship.

Acknowledgements

We would like to thank Ulrich Bauer for technical discussions on Ripser and Greg Henselman for discussions on Eirene. We also thank Greg Henselman, Matthew Kahle, and Cheng Xin on discussions about probability and apparent pairs. We acknowledge Birkan Gokbag for his help in developing Python bindings for Ripser++. We appreciate the constructive comments and suggestions of the anonymous reviewers. Finally, we are grateful for the insights and expert judgement in many discussions with Tamal Dey.

Cite AsGet BibTex

Simon Zhang, Mengbai Xiao, and Hao Wang. GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.70

Abstract

The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study the computational structure of Vietoris-Rips persistence barcodes, and identify several unique mathematical properties and algorithmic opportunities with connections to the GPU. Mathematically and empirically, we look into the properties of apparent pairs, which are independently identifiable persistence pairs comprising up to 99% of persistence pairs. We give theoretical upper and lower bounds of the apparent pair rate and model the average case. We also design massively parallel algorithms to take advantage of the very large number of simplices that can be processed independently of each other. Having identified these opportunities, we develop a GPU-accelerated software for computing Vietoris-Rips persistence barcodes, called Ripser++. The software achieves up to 30x speedup over the total execution time of the original Ripser and also reduces CPU-memory usage by up to 2.0x. We believe our GPU-acceleration based efforts open a new chapter for the advancement of topological data analysis in the post-Moore’s Law era.

Subject Classification

ACM Subject Classification
  • Theory of computation → Massively parallel algorithms
  • Software and its engineering → Massively parallel systems
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Parallel Algorithms
  • Topological Data Analysis
  • Vietoris-Rips
  • Persistent Homology
  • Apparent Pairs
  • High Performance Computing
  • GPU
  • Random Graphs

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