Statistical Inference with Reproducing Kernel Hilbert Space
2008, Statistical Learning Theory II
The Graduate University for Advanced Studies, ISM
Kenji Fukumizu(Institute of Statistical Mathematics)
Schedule: April.18 - Friday, 10:30-12:00
Place: (253(B) Kensyuu-shitsu, 2F)
We have class on April 18, 25, May 2, 9, 23, 30, June 13, 20, 27. July 25, September 12, 26
No class on May 16, June 6, July 4, 11, 18.
Purpose of course
An introduction to the methodology of statistical inferrence with positive definite
kernels or reproducing kernel Hilbert spaces. The course explains the basic
ideas of the methods with explanation of necessary mathematical background.
Various kernel methods are explained, with special focus on SVM. Also, recent development
of dependence anlaysis with kernels are explained in detail.
Plan of lectures
0. Outline and Information on the course
1. Introduction: overview of kernel methods
Basic idea of kernel method
Examples of kernel methods
2. Elements of positive definite kernel and reproducing kernel
Hilbert space (slides)
Definition and properties of positive definite kernel
Quick introduction to Hilbert spaces
Reproducing kernel Hilbert spaces
3. Methods with kernels (slides)
Kernel PCA, Kernel CCA, Kernel FDA, SVM
4. Support vector machines I
A quick course on convex optimization
Optimization in learning of SVM
5. Support vector machines II (slides)
Variants of SVM
6. Generalization ability of SVM (slides)
Basics of computational learning theory
generalization ability of SVM
7. Theory of positive definite kernel and reproducing kernel
Hilbert space (slides)
Negative definite kernel and Sch"onberg's theorem
Various examples of positive definite kernels
8. Mean Element in RKHS (slides)
Mean in RKHS
Homogeneity test with kernels
9. Dependence analysis with kernels (slides)
Dependence and independence with kernels
Conditional independence with kernels
Application to dimension reduction
10. Various aspects of kernel methods (slides)
Relation to Gaussian process
Relation to smoothing spline
Functional data alalysis
- Learning with Kernels. B.Schoelkopf and A.Smola. (2001) MIT Press.
- An Introduction to Support Vector Machines and Other Kernel-based Learning Methods.
N.Cristianini and J.Shawe-Taylor (2000) Cambridge University Press.
Report topics will be assgined during the course.
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