Many problems in computational biology and chemistry can be formalized as classical statistical problems, e.g., pattern recognition, regression or dimension reduction, with the caveat that the data are often not vectors. Indeed objects such as gene sequences, small molecules, protein 3D structures or phylogenetic trees, to name just a few, have particular structures which contain relevant information for the statistical problem but can hardly be encoded into finite-dimensional vector representations.
Kernel methods are a class of algorithms well suited for such problems. Indeed they extend the applicability of many statistical methods initially designed for vectors to virtually any type of data, without the need for explicit vectorization of the data. The price to pay for this extension to non-vectors is the need to define a positive definite kernel between the objects, formally equivalent to an implicit vectorization of the data. The "art" of kernel design for various objects have witnessed important advances in recent years, resulting in many state-of-the-art algorithms in computational biology and chemistry, as well as many other fields.
The goal of this course is to present the mathematical foundations of kernel methods, as well as the main approaches that have emerged so far in kernel design. The relevance of these methods will be illustrated by several examples in computational biology and chemistry.
Lecture take place usually in room C103, 1pm-3:30pm.
Homeworks are due at the begining of the following lecture, by hard copy at or (better) e-mail to Jean-Philippe.Vert@ensmp.fr. Implementations can be done in the programming language of your choice, e.g., the free R language, or Matlab and its free clone Octave
| Date | Topic | Slides | Homework | Data | |
| Jan 16, 2008 | Positive definite Kernels, Mercer theorem | 1-32 | Homework 1 | ||
| Feb 6, 2008 | RKHS | 33-44 | Homework 2 | ||
| Feb 15, 2008 | Kernel trick, first kernel algorithms | 45-65 | No homework! | ||
| Feb 20, 2008 | Representer theorem, kernel PCA, kernel ridge regression | 66-97 | Homework 3 | xtrain.txt, ytrain.txt, xtest.txt, ytest.txt | |
| Mar 5, 2008 | Pattern recognition, SVM | 98-127 | Homework 4 | data.txt,labels.txt | |
| Mar 12, 2008 | SVM, Kernel examples | 128-170 | Homework 5 | ||
| Mar 19, 2008 | Semigroup kernels | 171-189 | Homework 6 | ||
| Mar 26, 2008 | Kernels on graphs | 284-331 |