The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm

TL;DR

This paper introduces the distilled Vietoris Rips filtration, a memory-efficient and parallelisable method for computing persistent homology, significantly reducing computational resources needed for large point cloud data.

Contribution

It proposes a novel distilled Vietoris Rips filtration construction with proven isomorphism to standard persistent homology, and an efficient algorithm suitable for large datasets.

Findings

Memory footprint is significantly reduced.

Algorithm is highly parallelisable.

Applicable to any metric space with pairwise distances.

Abstract

The long computational time and large memory requirements for computing Vietoris Rips persistent homology from point clouds remains a significant deterrent to its application to big data. This paper aims to reduce the memory footprint of these computations. It presents a new construction, the distilled Vietoris Rips filtration, and proves that its persistent homology is isomorphic to that of standard Vietoris Rips. The distilled complex is constructed using a discrete Morse vector field defined on the reduced Vietoris Rips complex. The algorithm for building and reducing the distilled filtration boundary matrix is highly parallelisable and memory efficient. It can be implemented for point clouds in any metric space given the pairwise distance matrix.