Abstract
We present our recent work on both linear and nonlinear data reduction methods and algorithms: for the linear case we discuss results on structure analysis of SVD of columnpartitioned matrices and sparse low-rank approximation; for the nonlinear case we investigate methods for nonlinear dimensionality reduction and manifold learning. The problems we address have attracted great deal of interest in data mining and machine learning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bishop, M., Svensen, M., Williams, K. I., GTM: the generative topographie map**, Neural Computation, 1998, 10: 215–234.
Preedman, D., Efficient simplicial reconstructions of manifolds from their samples, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24: 1349–1357.
Hinton, G., Roweis, S., Stochastic neighbor embedding, Neural Information Processing Systems, 2003, 15: 833–840.
Kohonen, T., Self-organizing Maps, 3rd ed., Berlin: Springer-Verlag, 2000.
Ramsay, J. Silverman, W., Applied Functional Data Analysis, Berlin: Springer-Verlag, 2002.
Roweis, S., Saul, L., Nonlinear dimensionality reduction by locally linear embedding, Science, 2000, 290: 2323–2326.
Tenenbaum, J., De Silva, V., Langford, J., A global geometric framework for nonlinear dimension reduction, Science, 2000, 290: 2319–2323.
Xu, G., Kailath, T., Fast subspace decompsotion, IEEE Transactions on Signal Processing, 1994, 42: 539–551.
Xu, G., Zha, H., Golub, G. et al., Fast algorithms for updating signal subspaces, IEEE Transactions on Circuits and Systems, 1994, 41: 537–549.
Zha, H., Marques, O., Simon, H., Large-scale SVD and subspace-based methods for information retrieval, Proceedings of Irregular ’98, Lecture Notes in Computer Science, 1998, 1457: 29–42.
Zhang, Z., Zha, H., Structure and perturbation analysis of truncated SVDs for column-partitioned matrices, SIAM Journal on Matrix Analysis and Applications, 2001, 22: 1245–1262.
Zhang, Z., Zha, H., Simon, H., Low-rank approximations with sparse factors I: basic algorithms and error analysis, SIAM Journal on Matrix Analysis and Applications, 2002, 23: 706–727.
Stewart, G. W., Four algorithms for the efficient computation of truncated pivoted QR approximation to a sparse matrix, Numerische Mathematik, 1999, 83: 313–323.
Golub, G., Van Loan, C., Matrix Computations, 3nd ed., Baltimore, Maryland: Johns Hopkins University Press, 1996.
Teh, Y., Roweis, S., Automatic alignment of hidden representations, Neural Information Processing Systems, 2002, 15: 841–848.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Z., Zha, H. Linear low-rank approximation and nonlinear dimensionality reduction. Sci. China Ser. A-Math. 47, 908–920 (2004). https://doi.org/10.1360/04ys0016
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1360/04ys0016