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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., Dongarra, J.J., Croz, J.D., Hammarling, S., Greenbaum, A., McKenney, A., Sorensen, D.: LAPACK Users’ guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Tech. Rep. TR-2002-01, The University of Chicago CS (January 2002)
Bengio, Y., Paiement, J., Vincent, P.: Out-of-Sample extensions for LLE, isomap, MDS, eigenmaps, and spectral clustering. In: Advances in Neural Information Processing Systems, pp. 177–184. MIT Press (2003)
Bishop, C.M., James, G.D.: Analysis of multiphase flows using dual-energy gamma densitometry and neural networks. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 327(2-3), 580–593 (1993)
Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numerica 13 (2004)
Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the nystrom method. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 214–225 (2004)
Garcke, J., Griebel, M., Thess, M.: Data mining with sparse grids. Computing 67(3), 225–253 (2001)
He, X., Yan, S., Hu, Y., Zhang, H.: Learning a locality preserving subspace for visual recognition. In: Ninth IEEE International Conference on Computer Vision, vol. 1, pp. 385–392 (2003)
Heinecke, A., Pflüger, D.: Multi- and many-core data mining with adaptive sparse grids. In: Proceedings of the 2011 ACM International Conference on Computing Frontiers (May 2011)
Huang, D., Zhang, X., Huang, G., Pang, Y., Zhang, L., Liu, Z., Yu, N., Li, H.: Neighborhood Preserving Projections (NPP): a Novel Linear Dimension Reduction Method. In: Huang, D.-S., Zhang, X.-P., Huang, G.-B. (eds.) ICIC 2005. LNCS, vol. 3644, pp. 117–125. Springer, Heidelberg (2005)
Hubert, L., Arabie, P.: Comparing partitions. Journal of Classification 2(1), 193–218 (1985)
von Luxburg, U.: A tutorial on spectral clustering. Statistics and Computing 17, 395–416 (2007)
Pflüger, D.: Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dissertation, Institut für Informatik, Technische Universität München, München (February 2010)
Pflüger, D., Peherstorfer, B., Bungartz, H.J.: Spatially adaptive sparse grids for high-dimensional data-driven problems. Journal of Complexity 26(5), 508–522 (2010)
Qiao, H., Zhang, P., Wang, D., Zhang, B.: An explicit nonlinear mapping for manifold learning. CoRR abs/1001.2605 (2010)
Schraufstetter, S., Benk, J.: A general pricing technique based on theta-calculus and sparse grids. In: Proceedings of the ENUMATH 2009 Conference, Uppsala (December 2009)
Williams, C., Seeger, M.: Using the nyström method to speed up kernel machines. In: Advances in Neural Information Processing Systems, vol. 13, pp. 682–688. MIT Press (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Peherstorfer, B., Pflüger, D., Bungartz, HJ. (2011). A Sparse-Grid-Based Out-of-Sample Extension for Dimensionality Reduction and Clustering with Laplacian Eigenmaps. In: Wang, D., Reynolds, M. (eds) AI 2011: Advances in Artificial Intelligence. AI 2011. Lecture Notes in Computer Science(), vol 7106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25832-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-25832-9_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25831-2
Online ISBN: 978-3-642-25832-9
eBook Packages: Computer ScienceComputer Science (R0)