Topology-Preserving Dimensionality Reduction via Interleaving Optimization

TL;DR

This paper introduces a method that incorporates topological guarantees into dimensionality reduction by minimizing the interleaving distance between persistent homology, improving the preservation of data's topological features during embedding.

Contribution

It presents a novel optimization framework that explicitly minimizes the interleaving distance, ensuring topological correctness in dimensionality reduction, especially for linear projections.

Findings

Effective topological preservation demonstrated in data visualization

Optimization of linear projections based on interleaving distance

Framework applicable to various dimensionality reduction tasks

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.