Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering

Part of Advances in Neural Information Processing Systems 16 (NIPS 2003)

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Yoshua Bengio, Jean-françcois Paiement, Pascal Vincent, Olivier Delalleau, Nicolas Roux, Marie Ouimet


Several unsupervised learning algorithms based on an eigendecompo- sition provide either an embedding or a clustering only for given train- ing points, with no straightforward extension for out-of-sample examples short of recomputing eigenvectors. This paper provides a unified frame- work for extending Local Linear Embedding (LLE), Isomap, Laplacian Eigenmaps, Multi-Dimensional Scaling (for dimensionality reduction) as well as for Spectral Clustering. This framework is based on seeing these algorithms as learning eigenfunctions of a data-dependent kernel. Numerical experiments show that the generalizations performed have a level of error comparable to the variability of the embedding algorithms due to the choice of training data.