Linear-Size Approximations to the Vietoris-Rips Filtration

TL;DR

This paper introduces a method to construct a linear-size filtered simplicial complex that approximates the Vietoris-Rips filtration's persistence diagram efficiently, enabling scalable topological data analysis for large datasets.

Contribution

It presents the first construction of an O(n)-size filtered complex that approximates the Vietoris-Rips persistence diagram with a scalable, efficient algorithm based on hierarchical net-trees.

Findings

Constructed an O(n)-size filtered complex with good approximation guarantees.

Achieved $O(n \,\log n)$ construction time depending on doubling dimension.

Enabled scalable computation of persistence diagrams for large datasets.

Abstract

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees.