Semi-Supervised Learning on Riemannian Manifolds

We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. The central idea of our approach is that classification functions are naturally defined only on the submanifold in question rather than the total ambient space. Using the Laplace-Beltrami operator one produces a basis (the Laplacian Eigenmaps) for a Hilbert space of square integrable functions on the submanifold. To recover such a basis, only unlabeled examples are required. Once such a basis is obtained, training can be performed using the labeled data set.