Machine learning with kernel methods, Spring 2020

Julien Mairal and Jean-Philippe Vert

MSc Mathematics, Vision, Learning (MVA) (ENS Cachan)

MSc Data science (Ecole Polytechnique, ENSAE ParisTech, Telecom ParisTech, University Paris Saclay)

MSc Mathematics for Life Sciences (MathSV) (University Paris South, Ecole Polytechnique, ENS Cachan)

MSc Mathematiques, Apprentissage et Sciences Humaines (MASH) (PSL Research University, Paris-Dauphine University, ENS Paris)

Homework 1: here is the list of homeworks we received, as of Feb 27. If you don't see your name, please upload your homework again or contact us


Slides are frequently updated. Please let us know if you spot typos!


Many problems in real-world applications of machine learning 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 of numbers. For example, protein sequences and structures in computational biology, text and XML documents in web mining, segmented pictures in image processing, or time series in speech recognition and finance, 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 so-called positive definite kernel function 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 and successful applications in many domains.

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. We will start with a presentation of the theory of positive definite kernels and reproducing kernel Hilbert spaces, which will allow us to introduce several kernel methods including kernel principal component analysis and support vector machines. Then we will come back to the problem of defining the kernel. We will present the main results about Mercer kernels and semigroup kernels, as well as a few examples of kernel for strings and graphs, taken from applications in computational biology, text processing and image analysis. Finally we will touch upon topics of active research, such as large-scale kernel methods and deep kernel machines.



Lecture (in english) usually take place at MINES ParisTech, 60 boulevard Saint-Michel, 75006 Paris (RER Luxembourg), 1:30-4pm.

Please register to the class by filling in the following form

DatePlaceLecturerTopicSlidesExercicesMore material
Jan 15Mines L108JPVKernels, RKHS1-411, 2, 3, 5Uniqueness of the RKHS Aronszajn's theorem
Jan 22Mines V107JM Kernel tricks, Representer theorem42-82end of 5, 8
Jan 29Mines L118JPVKernel ridge and logistic regression83-11414-15
Feb 5Criteo, 32 Rue Blanche, 75009 Paris (register here)JMLarge-margin classifiers, SVMs115-165Homework (deadline Feb 26)Bartlett et al. 2003
Feb 12Break
Feb 19Break
Feb 26ENS Cachan, amphi Marie CurieJMUnsupervised learning, kernel PCA, K-means, CCA166-20211, 13
Mar 4ENS Cachan, amphi Marie CurieJPVGreen, Mercer, Herglotz, Bochner and friends203-27819, 22
Mar 11ENS Cachan, amphi Marie CurieJMLarge-scale learning with kernels, Deep kernel machines545-634
Mar 18CanceledJPV
Mar 25CanceledFinal exam


The final note will be an average of a homework (40%) and a final written exam (60%) based on a homework.

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