On the effectiveness of persistent homology

Abstract

Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. While 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 on which 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.

Topological data analysis of protein structure and inter/intra-molecular interaction changes attributable to amino acid mutations

“…The persistent homology method is independent of absolute coordinates and captures structural feature points according to the positional relationships (coordinates etc.) of the data points of interest; it is considered to be the best method for considering the features of systems in which the absolute coordinates of atoms change over time, as in this case [11] . Using this process, we obtain a time series of persistent diagrams, which plot the structural feature points for the Control and Case structures, respectively.…”

Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?

“…Another crucial fact that makes PH useful is its stability with respect to perturbations making KP robust to noise [19] mitigating the issues due to the open-world problem. Thus we use PH due to its effectiveness for limited resources and noise [50]. Concretely, the following are our key contributions:…”