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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, machine learning, numerical mathematics etc. The main emphases is on the development of a common set of tools that has proved to be useful in a wide range of applications in different areas. Topics will include concentration of measure, random matrices, Stein's methods and limit theorem of probability theory.
Program
- Introduction to concentration of measure; tensorization of variance.
- Chernoff estimate, Heffding's inequality; applications to multi-armed bandits; exploration vs exploitation; optimism in the face of uncertainty.
- Bernstein's inequality.
- Subgaussian and subexponential random variables.
- Concentration on the sphere and Gaussian concentration; Johnson-Lindenstrauss lemma.
- Matrix Bernstein's inequality.
- Applications to community detection and randomized algorithms in computational mathematics; estimation of covariance matrices and projectors.
- Poincaré's inequality and convergence of Markov processes; applications to diffusion based MCMC algorithms.
- Stein's method.
The Course is part of the International Thematic Program “Mathematical Foundations of Artificial Intelligence".
 RSS: Forthcoming seminars
 Lecturer
Naumov Aleksei Aleksandrovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
