Persistent Homology for the Evaluation of Dimensionality Reduction Schemes

Abstract

High-dimensional data sets are a prevalent occurrence in many application domains. This data is commonly visualized using dimensionality reduction (DR) methods. DR methods provide e.g. a two-dimensional embedding of the abstract data that retains relevant high-dimensional characteristics such as local distances between data points. Since the amount of DR algorithms from which users may choose is steadily increasing, assessing their quality becomes more and more important. We present a novel technique to quantify and compare the quality of DR algorithms that is based on persistent homology. An inherent beneficial property of persistent homology is its robustness against noise which makes it well suited for real world data. Our pipeline informs about the best DR technique for a given data set and chosen metric (e.g. preservation of local distances) and provides knowledge about the local quality of an embedding, thereby helping users understand the shortcomings of the selected DR method. The utility of our method is demonstrated using application data from multiple domains and a variety of commonly used DR methods.