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A Sparse-Grid-Based Out-of-Sample Extension for Dimensionality Reduction and Clustering with Laplacian Eigenmaps

  • Benjamin Peherstorfer
  • Dirk Pflüger
  • Hans-Joachim Bungartz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7106)

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.

Keywords

spectral methods manifold learning clustering sparse grids 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Benjamin Peherstorfer
    • 1
  • Dirk Pflüger
    • 1
  • Hans-Joachim Bungartz
    • 1
  1. 1.Department of InformaticsTechnische Universität MünchenGarchingGermany

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