In this paper we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). Assigning the Vietoris-Rips weights to simplices, DR experiences speed-up in the persistence calculations by only considering simplices that appear in the Delaunay triangulation of the point cloud. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR for an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

翻译：本文定义、实现并研究了一种用于计算欧几里得点云数据持续同调的单纯复形构造方法，称之为Delaunay-Rips复形（DR）。通过为单纯形赋予Vietoris-Rips权值，DR仅考虑出现在点云Delaunay三角剖分中的单纯形，从而加速持续同调计算。我们记录并比较了DR的Python实现与其他生成持续图的单纯复形构造方法。通过向点云数据施加充分条件，我们从理论上证明了DR生成的持续图的稳定性。当点云的Delaunay三角剖分在点扰动下发生改变时，我们证明DR生成的持续图会出现不稳定性。由于无法保证现实世界数据满足稳定性条件，我们通过机器学习应用中的实验比较，展示了DR相对于其他单纯复形在持续同调中的实际鲁棒性。实验发现，在ML-TDA流程中使用DR与其他单纯复形构造方法相比具有相当的性能。