Abstract
Kernel selection is one of the key issues both in recent research and application of kernel methods. This is usually done by minimizing either an estimate of generalization error or some other related performance measure. Use of notions of stability to estimate the generalization error has attracted much attention in recent years. Unfortunately, the existing notions of stability, proposed to derive the theoretical generalization error bounds, are difficult to be used for kernel selection in practice. It is well known that the kernel matrix contains most of the information needed by kernel methods, and the eigenvalues play an important role in the kernel matrix. Therefore, we aim at introducing a new notion of stability, called the spectral perturbation stability, to study the kernel selection problem. This proposed stability quantifies the spectral perturbation of the kernel matrix with respect to the changes in the training set. We establish the connection between the spectral perturbation stability and the generalization error. By minimizing the derived generalization error bound, we propose a new kernel selection criterion that can guarantee good generalization properties. In our criterion, the perturbation of the eigenvalues of the kernel matrix is efficiently computed by solving the derivative of a newly defined generalized kernel matrix. Both theoretical analysis and experimental results demonstrate that our criterion is sound and effective.
Similar content being viewed by others
References
Vapnik V. The Nature of Statistical Learning Theory. New York: Springer, 2000
Xu C, Peng Z M, Jing W F. Sparse kernel logistic regression based on 1/2 regularization. Sci China Inf Sci, 2013, 56: 042308
Chapelle O, Vapnik V, Bousquet O, et al. Choosing multiple parameters for support vector machines. Mach Learn, 2002, 46: 131–159
Xu Z B, Dai M, Meng D Y. Fast and efficient strategies for model selection of Gaussian support vector machine. IEEE Trans Syst Man Cybern B Cybern, 2009, 39: 1292–1307
Li G Z, Zhao R W, Qu H N, et al. Model selection for partial least squares based dimension reduction. Pattern Recognit Lett, 2012, 33: 524–529
Bartlett P, Mendelson S. Rademacher and Gaussian complexities: Risk bounds and structural results. J Mach Learn Res, 2002, 3: 463–482
Zhang T. Covering number bounds of certain regularized linear function classes. J Mach Learn Res, 2002, 2: 527–550
Zou B, Peng Z M, Xu Z B. The learning performance of support vector machine classification based on markov sampling. Sci China Inf Sci, 2013, 56: 032110
Schölkopf B, Smola A. Learning with Kernels. London: MIT Press, 2002
Luxburg U, Bousquet O, Schölkopf B. A compression approach to support vector model selection. J Mach Learn Res, 2004, 5: 293–323
Liu Y, Jiang S L, Liao S Z. Eigenvalues perturbation of integral operator for kernel selection. In: Proceedings of the 22nd ACM International Conference on Information and Knowledge Management, San Francisco, 2013. 2189–2198
Liu Y, Liao S Z, Hou Y X. Learning kernels with upper bounds of leave-one-out error. In: Proceedings of the 20th ACM International Conference on Information and Knowledge Management, Glasgow, 2011. 2205–2208
Debruyne M, Hubert M, Suykens J. Model selection in kernel based regression using the influence function. J Mach Learn Res, 2008, 9: 2377–2400
Liu Y, Jiang S L, Liao S Z. Efficient approximation of cross-validation for kernel methods using Bouligand influence function. In: Proceedings of the 31st International Conference on Machine Learning, Beijing, 2014. 324–332
Ding L Z, Liao S Z. Nyström approximate model selection for LSSVM. In: Proceedings of the 16th Pacific-Asia Conference on Knowledge Discovery and Data Mining, Kuala Lumpur, 2012. 282–293
Ding L Z, Liao S Z. Approximate model selection for large scale LSSVM. In: Proceedings of the 3rd Asian Conference on Machine Learning, Taoyuan, 2011. 165–180
Cristianini N, Shawe-Taylor J, Elisseeff A, et al. On kernel-target alignment. In: Proceedings of 2001 Neural Information Processing Systems Conference, Vancouver, 2001. 367–373
Cortes C, Mohri M, Rostamizadeh A. Two-stage learning kernel algorithms. In: Proceedings of the 27th International Conference on Machine Learning, Haifa, 2010. 239–246
Nguyen C H, Ho T B. An efficient kernel matrix evaluation measure. Pattern Recognit, 2008, 41: 3366–3372
Rogers W, Wagner T. A finite sample distribution-free performance bound for local discrimination rules. Ann Stat, 1978, 6: 506–514
Kearns M, Ron D. Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Comput, 1999, 11: 1472–1453
Bousquet O, Elisseeff A. Stability and generalization. J Mach Learn Res, 2002, 2: 499–526
Kutin S, Niyogi P. Almost-everywhere algorithmic stability and generalization error. In: Proceedings of the 18th Conference in Uncertainty in Artificial Intelligence, Alberta, 2002. 275–282
Poggio T, Rifkin R, Mukherjee S, et al. General conditions for predictivity in learning theory. Nature, 2004, 428: 419–422
Cortes C, Mohri M, Pechyony D, et al. Stability of transductive regression algorithms. In: Proceedings of the 25th International Conference on Machine Learning, Helsinki, 2008. 176–183
Shalev-Shwartz S, Shamir O, Srebro N, et al. Learnability, stability and uniform convergence. J Mach Learn Res, 2010, 11: 2635–2670
Cortes C, Mohri M, Talwalkar A. On the impact of kernel approximation on learning accuracy. In: Proceeding of the International Conference on Artificial Intelligence and Statistics, Sardinia, 2010. 113–120
Jiang Y, Ren J T. Eigenvalue sensitive feature selection. In: Proceedings of the 28th International Conference on Machine Learning, Washington, 2011. 89–96
Nguyen C H, Ho T B. Kernel matrix evaluation. In: Proceedings of the 20th International Joint Conference on Artifficial Intelligence, Hyderabad, 2007. 987–992
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, Y., Liao, S. Kernel selection with spectral perturbation stability of kernel matrix. Sci. China Inf. Sci. 57, 1–10 (2014). https://doi.org/10.1007/s11432-014-5090-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-014-5090-z