Topology-Preserving Dimensionality Reduction via Interleaving Optimization

Abstract

Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in finding an optimal linear projection. We demonstrate the utility of this framework to data visualization.

Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science

“…In contrast to dissimilarity-based projections, TopoMap is based on the non-differentiable 0-homology topological persistence evolution of cycles in the simplicial complexes resulting from a Euclidean distance-based Rips filtration [85]. Given a set of m-dimensional data instances P, TopoMap builds a complete weighted graph over P, weighting each edge (p i , p j ) with the Euclidean distance between the corresponding endpoints, d(p i , p j ).…”