  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.
