## Reproducing kernel Hilbert spaces in Machine Learning## Arthur Gretton (with Heiko Strathmann and Wittawat Jitkrittum) |
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All lecture locations are listed on p. 4 of the first set of slides.

Course announcements will be posted on the **mailing list**.

This page will contain slides and detailed notes for the kernel part of the course. The assignment may also be found here (at the bottom of the page). Note that the slides will be updated as the course progresses, and I modify them to answer questions I get in the classes. I'll put the date of last update next to each document - be sure to get the latest one. Let me know if you find errors.

There are sets of practice exercises and solutions further down the page (after the slides).

For questions on the course material, please email Heiko Strathmann or Wittawat Jitkrittum.

- Definition of a kernel, how it relates to a feature space
- Combining kernels to make new kernels
- The reproducing kernel Hilbert space
- Applications: difference in means, kernel PCA, kernel ridge regression

Lectures 4, 5, 6, 7, 8 **slides** and **notes**, last modified 25 Nov 2016

- Distance between means in RKHS, integral probability metrics, the maximum mean discrepancy (MMD), two-sample tests
- Choice of kernels for distinguishing distributions, characteristic kernels
- Covariance operator in RKHS: proof of existence, definition of norms (including HSIC, the Hilbert-Schmidt independence criterion)
- Application of HSIC to independence testing
- Application of HSIC to feature selection, taxonomy discovery.
- Introduction to independent component analysis, kernel ICA

Lecture 9 **slides** and **notes**, last modified 15 March 2016

- Introduction to convex optimization
- The representer theorem
- Large margin classification, support vector machines for clasification

Lecture 10 **Slides 1**, **Slides 2** , and **notes**, last modified 20 Mar 2013

- Metric, normed, and unitary spaces, Cauchy sequences and completion, Banach and Hilbert spaces
- Bounded linear operators and the Riesz Theorem
- Equivalent notions of an RKHS: existence of reproducing kernel, boundedness of the evaluation operator
- Positive definiteness of reproducing kernels, the Moore-Aronszajn Theorem
- Mercer's Theorem for representing kernels

Supplementary lecture **slides**, last modified 22 Mar 2012

- Loss and risk, estimation and approximation error, a new interpretation of MMD
- Why use an RKHS: comparison with other function classes (Lipschitz and bounded Lipschitz)
- Characteristic kernels and universal kernels