Topologically-Informed Atlas Learning

TL;DR

This paper introduces a novel atlas learning method that constructs topologically faithful embeddings of complex manifolds with holes, enabling better manifold learning in challenging topologies.

Contribution

It proposes a bottom-up atlas construction approach that preserves topological features, improving manifold embedding for data with complex topology.

Findings

Successfully constructs atlases for synthetic manifolds with holes

Learns human motion embeddings from motion capture data

Models articulated objects with accurate kinematic representations

Abstract

We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional Euclidean space by learning a coordinate chart, but it requires that the entire manifold can be embedded in a single chart. This is impossible for manifolds with holes. In such cases, it is necessary to learn an atlas: a collection of charts that collectively cover the entire manifold. We begin with many small charts, and combine them in a bottom-up approach, where charts are only combined if doing so will not introduce problematic topological features. When it is no longer possible to combine any charts, each chart is individually embedded with standard manifold learning techniques, completing the construction of the atlas. We show the efficacy of our method by constructing atlases for challenging synthetic manifolds; learning human motion embeddings from motion capture data; and learning kinematic models of articulated objects.