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Regular version of the site
Master 2019/2020

## High-dimensional Statistical Methods

Type: Compulsory course (Statistical Learning Theory)
Area of studies: Applied Mathematics and Informatics
When: 1 year, 3 module
Mode of studies: Full time
Instructors: Quentin Paris
Master’s programme: Statistical Learning Theory
Language: English
ECTS credits: 6

### Course Syllabus

#### Abstract

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. Prerequisite: Probability Theory. #### Learning Objectives

• Understand the effect of dimensionality on the performance of statistical methods
• Popular methods adapted to the high-dimensional setting #### Expected Learning Outcomes

• knowledge of what a sub-gaussian random variable is.
• Understanding the behaviour of suprema of random variables
• BIC, LASSO and SLOPE methods for high-dimensional linear regression
• Knowledge of basic probabilistic results related to random matrices and useful in statistics. #### Course Contents

• Сoncentration of sums of independent random variables
Subgaussian distributions; Subgamma distributions.
• Suprema
Finite case; Suprema over convex polytopes; Covering and packing numbers; Chaining bounds.
• High dimensional regression
BIC, LASSO and SLOPE estimators.
• Statistics and random matrices
Analysis and probability with matrices; Matrix version of Bernstein’s inequality; High dimensional PCA and random projections. #### Assessment Elements

• Home assignment 1
• Home assignment 2
• Final written test #### Interim Assessment

• Interim assessment (3 module)
0.2 * Final written test + 0.4 * Home assignment 1 + 0.4 * Home assignment 2 #### Recommended Core Bibliography

• Boucheron, S., Lugosi, G., Massart, P. Concentration inequalities: A nonasymptotic theory of independence. – Oxford university press, 2013.