Semi-supervised Learning Using an Unsupervised Atlas

In many machine learning problems, high-dimensional datasets often lie on or near manifolds of locally low-rank. This knowledge can be exploited to avoid the “curse of dimensionality” when learning a classifier. Explicit manifold learning formulations such as lle are rarely used for this purpose, and instead classifiers may make use of methods such as local co-ordinate coding or auto-encoders to implicitly characterise the manifold. We propose novel manifold-based kernels for semi-supervised and supervised learning. We show how smooth classifiers can be learnt from existing descriptions of manifolds that characterise the manifold as a set of piecewise affine charts, or an atlas. We experimentally validate the importance of this smoothness vs. the more natural piecewise smooth classifiers, and we show a significant improvement over competing methods on standard datasets. In the semi-supervised learning setting our experiments show how using unlabelled data to learn the detailed shape of the underlying manifold substantially improves the accuracy of a classifier trained on limited labelled data.