Cross-Matched Interval Prevalence of High Dimensional Point Clouds

Abstract

Topological Data Analysis (TDA) has been applied with success to solve problems across many scientific disciplines. However, in the setting of a point cloud $X$ sampled from a shape $\mathcal{S}$ of low intrinsic dimension embedded within high ambient dimension $\mathbb{R}^D$, persistent homology, a key element to many TDA pipelines, suffers from two problems. First, when relatively small amounts of noise are introduced to the point cloud, persistent homology is unable to recover the true shape of $\mathcal{S}$. Secondly, the computational complexity of persistent homology scales poorly with the size of a point cloud. Although there is recent work that addresses the first issue via topological bootstrapping methods and topological prevalence, these new techniques still fall victim to the second issue. Here we introduce the cross-matched prevalence image (CMPI), an image which approximates the topological prevalent information of said point cloud, requiring only computations of persistent homology on the scale of samples of the point cloud and not the entire point cloud itself. We compute the CMPI for high dimensional synthetic data, demonstrating that it performs similarly in noise robustness experiments and accurately captures prevalent topological features as compared to previous topological bootstrapping methods.