Abstract
A disadvantage of most nonlinear dimensionality reduction methods is that there are no explicit mappings to project high-dimensional features into low-dimensional representation space. Previously, some methods have been proposed to provide explicit mappings for nonlinear dimensionality reduction methods. Nevertheless, a disadvantage of these methods is that the learned mapping functions are combinations of all the original features, thus it is often difficult to interpret the results. In addition, the dense projection matrices of these approaches will cause a high cost of storage and computation. In this paper, a framework based on L1-norm regularization is presented to learn explicit sparse polynomial mappings for nonlinear dimensionality reduction. By using this framework and the method of locally linear embedding, we derive an explicit sparse nonlinear dimensionality reduction algorithm, which is named sparse neighborhood preserving polynomial embedding. Experimental results on real world classification and clustering problems demonstrate the effectiveness of our approach.
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
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15(6), 1373–1396 (2003)
Weinberger, K.Q., Saul, L.K.: Unsupervised learning of image manifolds by semidefinite programming. International Journal of Computer Vision 70(1), 77–90 (2006)
He, X., Niyogi, P.: Locality preserving projections. In: Advances in Neural Information Processing Systems, vol. 16, pp. 37–45. The MIT Press, Cambridge (2004)
Qiao, H., Zhang, P., Wang, D., Zhang, B.: An Explicit Nonlinear Mapping for Manifold Learning. IEEE Transactions on Cybernetics 43(1), 51–63 (2013)
Zhou, H., Hastie, T., Tibshirani, R.: Sparse principle component analysis. Journal of Computational and Graphical Statistics 15(2), 265–286 (2006)
Cai, D., He, X., Han, J.: Spectral regression: A unified approach for sparse subspace learning. In: Proceedings of the 7th IEEE International Conference on Data Mining, pp. 73–82 (2007)
Yan, S., Xu, D., Zhang, B., Zhang, H., Yang, Q., Lin, S.: Graph embedding and extensions: A general framework for dimensionality reduction. IEEE Transactions on Pattern Analysis and Machine Intelligence 24(1), 40–51 (2007)
Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B (Methodological), 267–288 (1996)
Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software 33(1), 1–22 (2010)
Martinez, A., Benavente, R.: The AR face database. CVC Tech. Report #24 (1998)
Sim, T., Baker, S., Bsat, M.: The CMU Pose, Illumination, and Expression Database. IEEE Transactions Pattern Analysis and Machine Intelligence 25(12), 1615–1618 (2003)
Jolliffe, I.: Principal component analysis. John Wiley & Sons, Ltd. (2005)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Xia, Y., Lu, Q., Feng, J., Bae, HY. (2014). An Explicit Sparse Mapping for Nonlinear Dimensionality Reduction. In: Miao, D., Pedrycz, W., Ślȩzak, D., Peters, G., Hu, Q., Wang, R. (eds) Rough Sets and Knowledge Technology. RSKT 2014. Lecture Notes in Computer Science(), vol 8818. Springer, Cham. https://doi.org/10.1007/978-3-319-11740-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-11740-9_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11739-3
Online ISBN: 978-3-319-11740-9
eBook Packages: Computer ScienceComputer Science (R0)