Predictive low-rank decomposition for kernel methods

ABSTRACT

Low-rank matrix decompositions are essential tools in the application of kernel methods to large-scale learning problems. These decompositions have generally been treated as black boxes---the decomposition of the kernel matrix that they deliver is independent of the specific learning task at hand---and this is a potentially significant source of inefficiency. In this paper, we present an algorithm that can exploit side information (e.g., classification labels, regression responses) in the computation of low-rank decompositions for kernel matrices. Our algorithm has the same favorable scaling as state-of-the-art methods such as incomplete Cholesky decomposition---it is linear in the number of data points and quadratic in the rank of the approximation. We present simulation results that show that our algorithm yields decompositions of significantly smaller rank than those found by incomplete Cholesky decomposition.

REFERENCES

• 1: Francis R. Bach , Michael I. Jordan, Kernel independent component analysis, The Journal of Machine Learning Research, 3, p.1-48, 3/1/2003
• 2: Shai Fine , Katya Scheinberg, Efficient svm training using low-rank kernel representations, The Journal of Machine Learning Research, 2, 3/1/2002
• 3: G. H. Golub and C. F. Van Loan. Matrix Computations. J. Hopkins Univ. Press, 1996.
• 4: T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer-Verlag, 2001.
• 5: Gert R. G. Lanckriet , Nello Cristianini , Peter Bartlett , Laurent El Ghaoui , Michael I. Jordan, Learning the Kernel Matrix with Semidefinite Programming, The Journal of Machine Learning Research, 5, p.27-72, 12/1/2004
• 6: B. Schölkopf and A. J. Smola. Learning with Kernels. MIT Press, 2001.
• 7: John Shawe-Taylor , Nello Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, New York, NY, 2004
• 8: Alex J. Smola , Bernhard Schökopf, Sparse Greedy Matrix Approximation for Machine Learning, Proceedings of the Seventeenth International Conference on Machine Learning, p.911-918, June 29-July 02, 2000
• 9: J. A. K. Suykens , J. Vandewalle, Least Squares Support Vector Machine Classifiers, Neural Processing Letters, v.9 n.3, p.293-300, June 1999 [doi>10.1023/A:1018628609742 ]
• 10: Christopher K. I. Williams , Matthias Seeger, The Effect of the Input Density Distribution on Kernel-based Classifiers, Proceedings of the Seventeenth International Conference on Machine Learning, p.1159-1166, June 29-July 02, 2000