Sparips

TL;DR

This paper introduces a new, easy-to-implement sparsification method for Rips filtrations using cover-trees, enabling scalable topological data analysis with explicit approximation guarantees and visualization tools.

Contribution

The authors propose a novel sparsification scheme based on cover-trees that improves scalability and provides explicit approximation guarantees for persistent homology computations.

Findings

The cover-tree based sparsification reduces computational complexity.

The method maintains similar topological features as the original Rips filtration.

An adapted visualization method effectively represents approximate persistence diagrams.

Abstract

Persistent homology of the Rips filtration allows to track topological features of a point cloud over scales, and is a foundational tool of topological data analysis. Unfortunately, the Rips-filtration is exponentially sized, when considered as a filtered simplicial complex. Hence, the computation of full persistence modules is impossible for all but the tiniest of datasets; when truncating the dimension of topological features, the situation becomes slightly less intractable, but still daunting for medium-sized datasets. It is theoretically possible to approximate the Rips-filtration by a much smaller and sparser, linear-sized simplicial complexs, however, possibly due to the complexity of existing approaches, we are not aware of any existing implementation. We propose a different sparsification scheme, based on cover-trees, that is easy to implement, while giving similar guarantees on the computational scaling. We further propose a visualization that is adapted to approximate persistence diagrams, by incorporating a variant of error bars and keeping track of all approximation guarantees, explicitly.