Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation
Statistical properties of kernel principal component analysis
Machine Learning
On the eigenspectrum of the gram matrix and the generalization error of kernel-PCA
IEEE Transactions on Information Theory
Functional Learning of Kernels for Information Fusion Purposes
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Semi-supervised Laplacian Regularization of Kernel Canonical Correlation Analysis
ECML PKDD '08 Proceedings of the 2008 European Conference on Machine Learning and Knowledge Discovery in Databases - Part I
On Relevant Dimensions in Kernel Feature Spaces
The Journal of Machine Learning Research
Accurate Probabilistic Error Bound for Eigenvalues of Kernel Matrix
ACML '09 Proceedings of the 1st Asian Conference on Machine Learning: Advances in Machine Learning
Kernel Analysis of Deep Networks
The Journal of Machine Learning Research
Eigenvalues perturbation of integral operator for kernel selection
Proceedings of the 22nd ACM international conference on Conference on information & knowledge management
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The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity. We derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration, reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations. Such scaling bounds have so far only been known for tail sums of eigenvalues. Asymptotically, the bounds presented here have a slower than stochastic rate, but the number of sample points necessary to make this disadvantage noticeable is often unrealistically large. Therefore, under practical conditions, and for all but the largest few eigenvalues, the bounds presented here form a significant improvement over existing non-scaling bounds.