Towards a Robust Persistence Diagram via Data-Dependent Kernel

Abstract

Topological Data Analysis (TDA) is used to extract topological features such as rings from point clouds. Recent works have identified that existing methods, which construct persistence diagrams in TDA, are not robust to noise and varied densities in a point cloud. This causes these methods to obtain incorrect topological features. We analyze the necessary properties of an approach that can address these two issues, and propose a new filter function for TDA based on a new data-dependent kernel that possesses these properties. Our empirical evaluation reveals that (i) the proposed kernel provides a better mean for UMAP dimensionality reduction (ii) the proposed filter function can significantly improve the performance of Topological Point Cloud Clustering (iii) the proposed filter function is a more effective way of constructing Persistence Diagram for t-SNE visualization and SVM classification than three existing methods of TDA, In addition, we explore the proposed filter’s performance on a more complex deformation named Riemannian stretching. Our proposed filter equipped with Sample Fermat distance outperforms all the other filters when noise and Riemannian stretching coexist. Code is available at https://github.com/IsolationKernel/Codes/tree/main/Lambda-kernel.