We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers
GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes
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.
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- Award ID(s):
- 1718450
- NSF-PAR ID:
- 10171713
- Date Published:
- Journal Name:
- Proceedings of Symposium of Computational Geometry (SoCG 2020)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract $$k\ge 0$$ $$n\ge 1$$ k Vietoris–Rips persistence diagrams ofall subsets of a given metric space with cardinality at mostn . We call these invariantspersistence sets and denote them as$${\textbf{D}}_{n,k}^{\textrm{VR}}$$ n andk , computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which$${\textbf{D}}_{4,1}^{\textrm{VR}}$$ X with cardinality$$2k+2$$ -
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