IsUMap: Manifold Learning and Data Visualization Leveraging Vietoris-Rips Filtrations

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 and metric realization of one-parameter filtrations of simplicial complexes. Inferring topological information from combinatorial models which have been built according to metric relations (Vietoris-Rips complexes) has proven useful in topological data analysis and general machine learning applications. This encourages the use of such objects for geometric inference. We extend this research direction by proposing a clear theoretical pipeline that not only provides a comprehensive guide for assigning a (triangulated) metric space to every admissible one- parameter filtration of simplicial complexes but also offers a method for merging these objects. With this, our method presents 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 with known geometries and in applications to data, in particular from computational biology.