[Paper Review] GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes

Simon Zhang, Mengbai Xiao+1 more|arXiv (Cornell University)|Jan 1, 2020

Topological and Geometric Data Analysis34 references18 citations

TL;DR

This paper presents Ripser++, a GPU-accelerated software for computing Vietoris-Rips persistence barcodes, leveraging apparent pairs and massively parallel algorithms to achieve up to 30x speedup over the original Ripser. It introduces theoretical bounds on apparent pairs and optimizes filtration construction, matrix reduction, and memory management for GPU execution, reducing CPU memory usage by up to 2.0x under favorable conditions.

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.

Motivation & Objective

• Address the high computational and memory demands of Vietoris-Rips persistence barcode computation in topological data analysis.
• Overcome sequential bottlenecks in matrix reduction and filtration construction that limit performance on CPUs.
• Leverage GPU hardware to exploit massive parallelism in simplex processing and apparent pair identification.
• Develop a theoretically grounded, high-performance software stack that scales efficiently under favorable sampling conditions.
• Enable faster, more memory-efficient TDA workflows for large-scale datasets in neuroscience, bioinformatics, and machine learning.

Proposed method

• Identify and exploit the mathematical property of 'apparent pairs'—persistence pairs that can be detected independently and comprise up to 99% of all pairs—enabling GPU-parallel detection.
• Design massively parallel algorithms for filtration construction with clearing and matrix reduction, using GPU-optimized data structures and memory access patterns.
• Implement a submatrix reduction strategy that tracks persistence pairs efficiently, reducing redundant computation and memory pressure.
• Use a hybrid approach: warp-based filtering for sparse cases and full sorting for dense cases, depending on the number of columns to be reduced.
• Apply theoretical bounds on apparent pair rates to guide algorithm design and validate empirical behavior.
• Integrate all components into Ripser++, an open-source GPU-accelerated software stack built on optimized CUDA kernels and memory management.

Experimental results

Research questions

• RQ1 What is the theoretical and empirical frequency of apparent pairs in Vietoris-Rips filtrations, and can they be exploited for GPU-parallel computation?
• RQ2 How can the computational bottlenecks in filtration construction and matrix reduction be restructured for massive GPU parallelism?
• RQ3 What are the expected work and depth complexities of a GPU-optimized Vietoris-Rips barcode computation under uniform sampling conditions?
• RQ4 To what extent can GPU acceleration reduce both execution time and CPU memory usage compared to CPU-only approaches?
• RQ5 Can the proposed methods be generalized beyond Vietoris-Rips filtrations to other types of filtrations such as flag or cubical complexes?

Key findings

• Apparent pairs constitute up to 99% of all persistence pairs in tested datasets, enabling massive parallelization opportunities.
• The paper proves tight upper and lower bounds on the apparent pair rate, with probabilistic lower bounds supporting high-frequency occurrence.
• Under uniform sampling, the expected work complexity of Ripser++ is near-linear in the number of simplices, with expected depth complexity dependent only on p-dimensional homological cycles.
• Ripser++ achieves up to 30x speedup over the original Ripser software on representative datasets, including real-world and synthetic point clouds.
• The software reduces CPU memory usage by up to 2.0x and requires at most 60% of the CPU memory used by Ripser, even when offloading computation to GPU device memory.
• Extensive experiments confirm that the empirical apparent pair rate closely matches theoretical upper bounds, validating the model’s predictive power.