- Lecturer: Zhongmin Qian
General prerequisites:
Part A Probability and Part A Integration are required.
Course term: Hilary
Course lecture information: 16 lectures
Course weight: 1
Course level: H
Assessment type: Written Examination
Course overview:
High-dimensional probability and high-dimensional statistics have emerged in recent years as ever more important topics due to the need to analyse vast amounts of complex data. The ideas and methods developed for dealing
with probability distributions on spaces of high-dimension (such as distributions of randomly sampled data with multiple attributes) have been used not only in pure mathematics but also in applications from stochastic simulation to statistics, data science to statistical mechanics.
This course will focus on the development of basic ideas and techniques such as elementary dimension-free tail estimates, concentration bounds, the metric entropy method, the Poincaré and logarithmic Sobolev inequalities, large deviation principles for rare events etc.
with probability distributions on spaces of high-dimension (such as distributions of randomly sampled data with multiple attributes) have been used not only in pure mathematics but also in applications from stochastic simulation to statistics, data science to statistical mechanics.
This course will focus on the development of basic ideas and techniques such as elementary dimension-free tail estimates, concentration bounds, the metric entropy method, the Poincaré and logarithmic Sobolev inequalities, large deviation principles for rare events etc.
Learning outcomes:
The students will learn the fundamental ideas and modern tools for
handling distributions on high-dimensional spaces, and understand
some special features of probability distributions on such spaces, for example
the concentration of probability laws on small regions of low dimensional
subspaces.
handling distributions on high-dimensional spaces, and understand
some special features of probability distributions on such spaces, for example
the concentration of probability laws on small regions of low dimensional
subspaces.
Course synopsis:
Derive a few elementary but important tail estimates for distributions in terms of moments, variances and other statistical characteristics.
Strong law of large numbers, Cramér's large deviation principle.
Elementary results on concentration of probabilities, concentration functions,
Heat semi-group and Ornstein-Uhlenbeck semi-group, Gaussian measures, Poincaré inequality, logarithmic Sobolev inequalities.
Concentration for Gaussian measures such as Borell inequality, the isoperimetric inequality for Gaussian measures.
Building the connection between concentration estimates and isoperimetric inequalities on Euclidean spaces and spheres, and indicating their significance in data science
Strong law of large numbers, Cramér's large deviation principle.
Elementary results on concentration of probabilities, concentration functions,
Heat semi-group and Ornstein-Uhlenbeck semi-group, Gaussian measures, Poincaré inequality, logarithmic Sobolev inequalities.
Concentration for Gaussian measures such as Borell inequality, the isoperimetric inequality for Gaussian measures.
Building the connection between concentration estimates and isoperimetric inequalities on Euclidean spaces and spheres, and indicating their significance in data science