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The eigenvalues of the kernel matrix play an important role in a number of kernel methods. It is well known that these eigenvalues converge as the number of samples tends to infinity. We derive a probabilistic finite sample size bound on the approximation error of an individual eigenvalue, which has the important property that the bound scales with the dominate eigenvalue under consideration, reflecting the accurate behavior of the approximation error as predicted by asymptotic results and observed in numerical simulations. Under practical conditions, the bound presented here forms a significant improvement over existing non-scaling bound. Applications of this theoretical finding in kernel matrix selection and kernel target alignment are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a generic diffusion kernel for graph-based semi-supervised learning, whose kernel matrix is generated with a Taylor expansion on the generating function of diffusion similarity matrix. The generic diffusion kernel subsumes common known diffusion kernels, and provides a kernel framework for semi-supervised learning. Specifically, we first present the definition of diffusion similarity matrix, and lay the theoretical foundation for our approach. Then we derive the 2- and d-diffusion kernels, and naturally extend them to the generic diffusion kernel. Further we prove that small eigenvalues of the generic diffusion kernel correspond to smooth eigenvectors over the graph. This property is critical for the construction of generic diffusion kernels. Experiments on simulated and benchmark databases demonstrate that the generic diffusion kernel is sound and effective. [ABSTRACT FROM AUTHOR]

Gaussian mean field is an important paradigm of cluster-based variational inference, and its cluster selection is critical to the tradeoff between the variational accuracy and the computational complexity of cluster-based variational inference. In this paper, we explore a coupling based cluster selection method for Gaussian mean fields. First, we propose the model coupling and the quasi-coupling concepts on Gaussian Markov random field, and prove the coupling-accuracy theorem for Gaussian mean fields, which regards the quasi-coupling as a cluster selection criterion. Then we design a normalized cluster selection algorithm based on the criterion for Gaussian mean fields. Finally, we design numerical experiments to demonstrate the validity and efficiency of the cluster selection method and algorithm. [ABSTRACT FROM AUTHOR]