New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices
Nima Reyhani, Hideitsu Hino, Ricardo Vigario
Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for other methods the spectral information can be used to estimate the excess risk. An important question remains on how close the sample eigenvalues/eigenvectors are to the population values. In this paper, we improve earlier results on concentration bounds for eigenvalues of general kernel matrices. For distance and inner product kernel functions, e.g. radial basis functions, we provide new concentration bounds, which are characterized by the eigenvalues of the sample covariance matrix. Meanwhile, the obstacles for sharper bounds are accounted for and partially addressed. As a case study, we derive a concentration inequality for sample kernel target-alignment.
PDF Link: /papers/11/p627-reyhani.pdf
AUTHOR = "Nima Reyhani
and Hideitsu Hino and Ricardo Vigario",
TITLE = "New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices",
BOOKTITLE = "Proceedings of the Twenty-Seventh Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-11)",
PUBLISHER = "AUAI Press",
ADDRESS = "Corvallis, Oregon",
YEAR = "2011",
PAGES = "627--634"