Scaffoldings and Spines: Organizing High-Dimensional Data Using Cover Trees, Local Principal Component Analysis, and Persistent Homology

Abstract

We propose a flexible and multi-scale method for organizing, visualizing, and understanding point cloud datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree for a dataset using an adaptive threshold that is based on multi-scale local principal component analysis. The resulting cover tree nodes reflect the local geometry of the space and are organized via a scaffolding graph. In the second part of the algorithm, the goals are to uncover the strata that make up the underlying stratified space using a local dimension estimation procedure and topological data analysis, as well as to ultimately visualize the results in a simplified spine graph. We demonstrate our technique on several synthetic examples and then use it to visualize song structure in musical audio data.