Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used in many different types of applications, the reasons behind its success remain elusive; in particular, it is not known for which classes of problems it is most effective, or to what extent it can detect geometric or topological features. The goal of this work is to identify some types of problems where PH performs well or even better than other methods in data analysis. We consider three fundamental shape analysis tasks: the detection of the number of holes, curvature and convexity from 2D and 3D point clouds sampled from shapes. Experiments demonstrate that PH is successful in these tasks, outperforming several baselines, including PointNet, an architecture inspired precisely by the properties of point clouds. In addition, we observe that PH remains effective for limited computational resources and limited training data, as well as out-of-distribution test data, including various data transformations and noise. For convexity detection, we provide a theoretical guarantee that PH is effective for this task, and demonstrate the detection of a convexity measure on the FLAVIA dataset of plant leaf images.
翻译:持久性同系物(PH)是地形数据分析中最受欢迎的方法之一。尽管PH被用于许多不同类型的应用,但其成功的原因仍然难以预料;特别是,不知道PH在哪些类别的问题上最为有效,或者它能够在多大程度上探测到几何或地貌特征。这项工作的目标是查明PH在数据分析方面表现良好甚至比其他方法更好的一些问题。我们考虑三项基本的形状分析任务:从2D和3D点取样的云层中探测出洞数、弯曲性和共性。实验表明PH在这些任务中取得了成功,超过了几个基线,包括点网,这是精确受点云特性影响的结构。此外,我们注意到PH对于有限的计算资源和有限的培训数据,以及包括各种数据转换和噪音在内的分配外测试数据仍然有效。关于固态检测,我们从理论上保证PH对这项任务有效,并展示了对FLAVIA图像的粘合度测量。