Bifunction and Interlevel Delaunay Trifiltrations

2026-05-20

AI Summary

This work proposes the first three-parameter Delaunay trifiltration for point clouds equipped with ℝ²-valued functions, satisfying weak topological equivalence to enable multiparameter persistent homology in time-varying data. Extending the classical Delaunay filtration to higher-dimensional function-valued settings, the method preserves weak equivalence with the offset filtration while introducing an efficient algorithm with time complexity O(|X|^{⌈d/2⌉+2}) and a computational framework whose memory usage grows nearly linearly with input size. Experimental results demonstrate that the approach effectively handles thousands of points in ℝ³, offering both scalability and practicality. This framework thus provides a computationally feasible new tool for multiparameter topological data analysis.

Abstract

A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

Problem

• Delaunay trifiltration
• multicover bifiltration
• topological data analysis
• weak equivalence
• multiparameter persistence

Innovation

• Delaunay trifiltration
• multiparameter persistence
• topological data analysis
• weak equivalence
• computational topology