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

Benjamin Peherstorfer, Dirk Pflüger, Hans Joachim Bungartz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationAI 2011
Subtitle of host publicationAdvances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings
Pages112-121
Number of pages10
DOIs
StatePublished - Dec 23 2011
Event24th Australasian Joint Conference on Artificial Intelligence, AI 2011 - Perth, WA, Australia
Duration: Dec 5 2011Dec 8 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7106 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other24th Australasian Joint Conference on Artificial Intelligence, AI 2011
CountryAustralia
CityPerth, WA
Period12/5/1112/8/11

Fingerprint

Sparse Grids
Dimensionality Reduction
Clustering
Manifold Learning
Laplace-Beltrami Operator
Sample point
Eigenvalues and eigenfunctions
Eigenfunctions
Graph in graph theory
Training

Keywords

  • clustering
  • manifold learning
  • sparse grids
  • spectral methods

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Peherstorfer, B., Pflüger, D., & Bungartz, H. J. (2011). A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. In AI 2011: Advances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings (pp. 112-121). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7106 LNAI). https://doi.org/10.1007/978-3-642-25832-9_12

A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. / Peherstorfer, Benjamin; Pflüger, Dirk; Bungartz, Hans Joachim.

AI 2011: Advances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings. 2011. p. 112-121 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7106 LNAI).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Peherstorfer, B, Pflüger, D & Bungartz, HJ 2011, A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. in AI 2011: Advances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7106 LNAI, pp. 112-121, 24th Australasian Joint Conference on Artificial Intelligence, AI 2011, Perth, WA, Australia, 12/5/11. https://doi.org/10.1007/978-3-642-25832-9_12
Peherstorfer B, Pflüger D, Bungartz HJ. A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. In AI 2011: Advances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings. 2011. p. 112-121. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-25832-9_12
Peherstorfer, Benjamin ; Pflüger, Dirk ; Bungartz, Hans Joachim. / A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. AI 2011: Advances in Artificial Intelligence - 24th Australasian Joint Conference, Proceedings. 2011. pp. 112-121 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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