A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with laplacian eigenmaps

Authors:
Benjamin Peherstorfer;Dirk Pflüger;Hans-Joachim Bungartz
Affiliations:
Department of Informatics, Technische Universität München, Garching, Germany;Department of Informatics, Technische Universität München, Garching, Germany;Department of Informatics, Technische Universität München, Garching, Germany
Venue:
AI'11 Proceedings of the 24th international conference on Advances in Artificial Intelligence
Year:
2011

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Abstract

Spectral graph theoretic methods such as Laplacian Eigenmaps are among the most popular algorithms for manifold learning and clustering. One drawback of these methods is, however, that they do not provide a natural out-of-sample extension. They only provide an embedding for the given training data. We propose to use sparse grid functions to approximate the eigenfunctions of the Laplace-Beltrami operator. We then have an explicit mapping between ambient and latent space. Thus, out-of-sample points can be mapped as well. We present results for synthetic and real-world examples to support the effectiveness of the sparse-grid-based explicit mapping.