A Sparse Delaunay Filtration

TL;DR

This paper introduces a sparse Delaunay filtration that approximates the persistence diagram of a point set's distance function in high dimensions, significantly reducing complexity while maintaining accuracy.

Contribution

It presents a novel sparse filtration method that uses only linear simplices to approximate the Delaunay complex, enabling efficient computation of persistence diagrams.

Findings

Uses only O(n) simplices instead of O(n^{ceil(d/2)})

Provides a simple proof of correctness based on geometric duality

Enables efficient construction of the filtration

Abstract

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in $R^d$. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size $O(n^{\lceil d/2 \rceil})$. In contrast, our construction uses only $O(n)$ simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in $d+1$ dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a $(d+1)$-dimensional Voronoi construction similar to the standard Delaunay filtration complex. We also, show how this complex can be efficiently constructed.