GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes
About
The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study its computational structure 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 prove tight upper and lower bounds of the apparent pair rate and some probabilistic lower bounds. 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++. Under nice sampling conditions, we show that the expected work complexity of our algorithm is near linear in the number of simplices. The expected depth complexity is dependent only on the computation of the expected number of $p$-dimensional homological cycles. 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.