Topology-Preserving Dimensionality Reduction 

Abstract— In an era where data is increasingly central across disciplines, effective visualization has become essential for interpreting complex datasets. Dimensionality reduction techniques are widely employed to project high-dimensional data into lower dimensions while preserving structural properties. In this paper, we extend manifold-landmarking-based dimensionality reduction to improve homological preservation, a key aspect of topology preservation. We first introduce AdaMapper, a mapper-based algorithm that leverages persistence diagrams to guide skeleton construction and landmark selection. AdaMapper is parameter-free by default and adaptively refines covers in regions where topological loops occur. We then propose AdaHIsomap, which combines the strengths of landmark Isomap with homology-informed landmarks and incorporates random anchor points to balance distance and homology preservation. We evaluate both methods on diverse datasets—including high-dimensional point clouds, scientific simulations, networks, and imaging—and benchmark them against state-of-the-art approaches.

Committee: Lin Yan (major professor), Ali Jannesari, and Mengdi Huai