Teaching
High Dimensional Statistics
The course presents an introduction to modern statistical and probabilistic methods for data analysis, emphasising finite sample guarantees and problems arising from high-dimensional data. The course is mathematically oriented and level of the material ranges from a solid undergraduate to a graduate level. Topics studied include for instance Concentration Inequalities, High Dimensional Linear Regression and Matrix estimation.
Themes:
- Theme 1. Concentration of random variables pdf
- Theme 2. Concentration of sums of independent random variables pdf
- Theme 3. Suprema of random variables pdf
- Theme 4. High-dimensional linear regression with the BIC pdf
- Theme 5. High-dimensional linear regression with the LASSO pdf
- Theme 6. Covariance matrix estimation (notes coming soon)
- Theme 7. High-dimensional PCA (notes coming soon)
- Theme 8. Concentration of sums of independent symmetric matrices pdf
- Theme 9. Community detection in the stochastic bloc model pdf
- Theme 10. Introduction to statistical learning theory pdf
- Theme 11. Empirical process theory and ERM pdf
- Theme 12. PAC bayesian learning pdf
References for the course:
- High-Dimensional Statistics. Lecture Notes. Rigollet P. and Hütter J-C.
- High-Dimensional Probability: An Introduction with Applications in Data Science (Second Edition) Vershynin R.
- Probability in High-Dimension van-Handel R.
- High-Dimensional Statistics: A Non-Asymptotic Viewpoint Wainwright M. J.
Optimal Transport
This course is an introduction to the theory of Optimal Transport, aiming at presenting some of the basic results in the field. The course is offered as an elective course in several Master and PhD programs at HSE University. Prerequisites for the course include a basic knowledge of Differential Calculus, Convex analysis, Measure Theory and Topology.
Lectures:
- Lecture 1. The optimal transport problem pdf
- Lecture 2. Existence of optimal transport plans pdf
- Lecture 3. Kantorovich-Wasserstein distances pdf
- Lecture 4. Necessary and sufficient optimality conditions pdf
- Lecture 5. Duality and existence of optimal transport maps pdf
- Lecture 6. Continuity equation (notes coming soon)
- Lecture 7. The formal Riemannian structure of W_2 (notes coming soon)
References for the course:
- Lectures on Optimal Transport. Ambrosio L., Brué E. and Semola D.
- Statistical Optimal Transport. Chewi S., Niles-Weed J. and Rigollet P.
- Задачи Монжа и Канторовича оптимальной транспортировки. Богачев В.И., Колесников А. В., Шапошников С. В. (in Russian)
- A user’s guide to optimal transport. Ambrosio L. and Gigli N.
- Optimal Transport: Old and New. C. Villani.
- Topics in Optimal Transportation. C. Villani.
- Optimal Transport for Applied Mathematicians. P. Santambrogio.
- Computational Optimal Transport With Applications to Data Science. Peyré G. and Cuturi M.