Master’s Programme “Statistical Learning Theory”, 2nd year
The aim of this course is to provide an introduction to asymptotic and non-asymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics.
Lectures: Alexey Naumov (naumovne@gmail.com).
Seminars: Leonid Iosipoi (iosipoileonid@gmail.com).
Grade Components: home assignments (40%) + individual project (20%) + oral final exam (40%).
Important dates: defence of projects – December 6, final exam – December 13.
Seminars
12.09.2019 |
Catalan numbers.Recursive Definition of the Catalan numbers (balanced parentheses, Dyck paths, rooted binary trees, rooted trees). Generating function and explicit formula for the Catalan numbers. Notes: Problem set 1. Wigner’s Semicircular law.Wigner matrices. Method of Moments. Moments of the Semicircular law. Proof of the Semicircular law (convergence in expectation). Generalized Wigner and Wigner type matrices. Notes: Problem set 2.
References |
20.09.2019 |
Marchenko-Pastur law.Sample covariance matrices. Moments of the Marchenko-Pastur law. Proof of the Marchenko-Pastur law (convergence in expectation). Notes: Problem set 3.
References |
27.09.2019 |
Basic concentration inequalities and apllications.Basic concentration inequalities: Markov and Chernoff bounds. Sub-Gaussian and sub-exponential random variables. Hoeffding bound. Johnson-Lindenstrauss embedding. Erdos-Renyi model. Notes: Problem set 4.
References |
04.10.2019 |
Epsilon-net argument.Covering number of a unit sphere. Epsilon-net argument. A bound on the operator norm of a matrix with sub-Gaussian entries. Optimality of the epsilon-net argument. Notes: Problem set 5.
References |
11.10.2019 |
Epsilon-net argument for sample covariance matrices.Sub-Gaussian random vectors. Epsilon-net argument for sample covariance matrices. Notes: Problem set 6.
References |
18.10.2019 |
Comparison Method. Part I.Lipschitz functions of Gaussian variables. Gaussian comparison inequalities (Sudakov-Fernique inequality). Bound on the expected value of the operator norm. Notes: Problem set 7.
References |
15.11.2019 |
Comparison Method. Part II.Lipschitz functions of Gaussian variables. Gaussian comparison inequalities (Sudakov-Fernique inequality). Bound on the expected value of the operator norm. Notes: Problem set 7.
References |
22.11.2019 |
Bernstein-type boundBernstein condition for random variables. Bernstein inequality for random variables. Bernstein condition for random matrices. Bernstein inequality for random matrices. Notes: Problem set 8.
References |