Master 2 courses in Mathematics - Statistics track
The Statistics speciality of the Master's degree aims to train statisticians through both theoretical and applied methods.
2nd year Master courses (M2) taught in English
Semester 1
2nd year Master - 1st Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
This course presents convergence results for many sequential algorithms of Machine Learning, both in the deterministic and random setting. It will be shown that sequential learning provides adaptive and robust solutions to many convex optimization problems, with and without constraints. Convergence of algorithms will be illustrated with R and Python on the MNIST dataset.
Prerequisites
Fundamentals of probability and statistics, scientific computation in Python or R.
Contact
Olivier Wintenberger (olivier.wintenberger@sorbonne-universite.fr)
Semester 2
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
This course presents the concepts of a priori law/a posteriori law and the general framework for obtaining convergence speeds. The course covers
Gaussian processes, Dirichlet processes and multiplicative cascades. A focus is also made on deep neural networks and deep-ReLU laws, deep Gaussian processes.
Prerequisites
Fundamentals of probability and statistics.
Contact
Ismaël Castillo (ismael.castillo@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
This course presents how to use neural networks for adaptive numerical approximation.
Topics covered:
- Functions that can be represented by neural networks
- Elementary proofs of Cybenko's theorem
- The Takagi function
- Construction of datasets and curse of dimension
- Interpretation of stochastic gradient algorithms in the form of ordinary differential equations
- Applications to problems from scientific computing for CFD in relation with image classification
Illustration with some software.
Prerequisites
Fundamentals of analysis and interest in programming.
Contact
Bruno Desprès (bruno.despres@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
The goal of this course is twofold: on the one hand, to discover the real challenges of basic biology and medicine where statistical learning is already successfully used; on the other hand, to acquire the basics for modeling complex medical data.
Prerequisites
Fundamentals of probability and statistics, linear algebra, Python.
Contact
Xavier Tannier (xavier.tannier@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 1,5 ECTS - English Level: Scientific English (no test required)
Brief Description
This lecture provides an overview of PAC-Bayesian theory, starting from statistical learning theory (generalization bounds and oracle inequalities) and covering algorithmic developments by variational inference, up to recent PAC-Bayesian analyses of generalization properties of deep neural networks.
Prerequisites
Fundamentals of probability and statistics.
Contact
Badr-Eddine Chérief-Abdellatif (badr-eddine.cherief-abdellatif@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
Simulating random variables in high dimension is a real challenge for many recent machine learning problems and for learning deep generative models. This problem is encountered, for example, in a Bayesian context when the a posteriori law is known to within one normalization constant, in the context of variational autoencoders or for the metamodeling of complex dynamic systems.
Many methods are based on "Importance Sampling" or "Sequential Monte Carlo" approaches, whose main elements we will recall. To overcome the weaknesses inherent to these methodologies in high dimension or for deep generative models (based on recurrent networks, dense or convolutional networks), we will study in this course recent solutions with emphasis on methodological aspects. The functioning of these methods will be illustrated using public datasets for computer vision and time series prediction problems.
Prerequisites
Fundamentals of probability and statistics.
Contact
Sylvain Le Corff (sylvain.le_corff@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
The objective of this course is twofold: to illustrate the processing of high-dimensional data when data is missing (through the prism of compressed acquisition and matrix completion), and to acquire the basics of convex optimization. These two topics, which will be addressed in concert as they are closely related, open the way to many other statistical learning domains and problems encountered in data science.
Prerequisites
Fundamentals of probability and statistics, linear algebra, Python.
Contact
Claire Boyer (claire.boyer@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
This course will attempt to provide an overview of the latest mathematical trends in the machine learning and statistical learning community.
Topics covered:
- Approximation theory for neural networks
- VC dimension for neural networks
- Minimal bounds for regression with neural networks
- GANs
- Implicit bias in gradient descent
- Interpolation & benign overfitting
- Privacy
Prerequisites
Fundamentals of probability and statistics, linear algebra.
Contact
Eddie Aamari (eddie.aamari@sorbonne-universite.fr)
2nd year Master - 2nd Semester - 3 ECTS - English Level: Scientific English (no test required)
Brief Description
Data can often be represented by point clouds in high-dimensional spaces. In practice, it is found that these points are not uniformly distributed in the surrounding space: they are located near non-linear structures of smaller dimension, such as curves or surfaces, which are interesting to understand. Geometric inference, also called topological data analysis, is a recent field consisting in the study of statistical aspects associated with the geometry of data. This course aims to provide an introduction to this rapidly growing field.
Prerequisites
Fundamentals of probability and statistics.
Contact
Eddie Aamari (eddie.aamari@sorbonne-universite.fr)