IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations

TL;DR

IsUMap is a new manifold learning method that combines UMAP, Isomap, and Vietoris-Rips filtrations to better capture complex data structures and local geometries, outperforming existing techniques.

Contribution

It introduces a systematic metric construction for locally distorted spaces, improving data representation in manifold learning.

Findings

Significant improvements in representation quality on benchmark datasets

Effective handling of non-uniform data distributions

Enhanced capture of local geometries

Abstract

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.