Learning a Manifold as an Atlas

Abstract

In this work, we return to the underlying mathematical definition of a manifold and directly characterise learning a manifold as finding an atlas, or a set of overlapping charts, that accurately describe local structure. We formulate the problem of learning the manifold as an optimisation that simultaneously refines the continuous parameters defining the charts, and the discrete assignment of points to charts. In contrast to existing methods, this direct formulation of a manifold does not require "unwrapping" the manifold into a lower dimensional space and allows us to learn closed manifolds of interest to vision, such as those corresponding to gait cycles or camera pose. We report state-ofthe-art results for manifold based nearest neighbour classification on vision datasets, and show how the same techniques can be applied to the 3D reconstruction of human motion from a single image.