IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations
Lukas Silvester Barth, Fatemeh, Fahimi, Parvaneh Joharinad, Jürgen Jost, Janis Keck
TL;DR
IsUMap addresses the challenge of producing informative, geometry-faithful low-dimensional representations for data with non-uniform distributions by merging locally distorted metrics derived from Vietoris-Rips filtrations into a global intrinsic metric via metric realization. It blends Isomap and UMAP within the framework of uber metric spaces, using star-graph local metrics, $t$-conorm merging, and shortest-path completion to generate a unified distance and a low-dimensional embedding via MDS. The work provides both theoretical foundations (weighted simplicial complexes, metric realization, and category-theoretic framing) and extensive empirical demonstrations across synthetic manifolds, image data, non-uniform hemispherical distributions, knotted proteins, and RNA-velocity trajectories, illustrating improved geometric fidelity and topological structure preservation. The approach offers a robust, interpretable pipeline for visualization and downstream tasks in scenarios with density variation and complex local geometry.
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.