Homology-Preserving Dimensionality Reduction via Adaptive Mapper and Landmark Isomap

2026-06-03

AI Summary

This work addresses the challenge of preserving essential topological structures—such as connected components and loops—during dimensionality reduction of high-dimensional data. To this end, the authors propose two novel methods, AdaMapper and AdaHIsomap, which integrate the Mapper framework, persistent homology analysis, and an enhanced landmark selection strategy to simultaneously maintain local distances and global homological features. The key innovation lies in an adaptive cover refinement mechanism guided by persistence diagrams and the incorporation of homological information into the landmark selection process of Landmark Isomap. Experimental results demonstrate that the proposed approaches significantly outperform existing techniques across multiple high-dimensional datasets, achieving markedly improved fidelity in preserving topological characteristics in the reduced embeddings.

Abstract

As data becomes increasingly central across engineering and scientific disciplines, effective visualization is essential for interpreting complex, high-dimensional structures. Dimensionality reduction techniques project high-dimensional data into lower dimensions while aiming to preserve structural properties such as pairwise distances and local neighborhoods. In this paper, we focus on improving homological preservation, that is, the retention of topological features such as connected components and loops, which is critical for maintaining global shape and continuity. We first introduce AdaMapper, a Mapper-based algorithm that leverages persistence diagrams to guide both skeleton construction and landmark selection. AdaMapper incorporates an adaptive refinement strategy that automatically increases cover resolution in regions exhibiting topological loops. We then propose AdaHIsomap, which extends landmark Isomap by incorporating homology-informed landmark selection and augmenting it with random anchor points to better balance distance and homology preservation. We evaluate both methods on a diverse set of datasets, including high-dimensional point clouds, scientific simulations, networks, and image data, and benchmark their performance against state-of-the-art approaches.

Problem

• homology preservation
• dimensionality reduction
• topological features
• Mapper
• Isomap

Innovation

• homology preservation
• adaptive Mapper
• landmark Isomap
• persistence diagrams
• topological data analysis