Title: Introduction to high-dimensional probability: some basic concentration inequalities and useful distributions
|Affiliation:||University of Waterloo|
Abstract: In this seminar, we introduce important tools from high-dimensional probability useful in studying applications in data science such as covariance estimation, matrix completion, elements of statistical learning, networks, semidefinite programming, and many more. High-dimensional probability is an area of probability theory that studies random objects in R^n where the dimension of n is large. We will learn, in the seminar, basic theoretical skills (e.g. concentration inequalities, sub-gaussian distributions) for analyzing random vectors, random matrices, and random projections. An outline of the seminar is as follows: (1) we will cover Chapters 2-6 from “High-dimensional probability” by Roman Vershynin (available online) and (2) we will present papers in data science which utilize these tools.
In this talk, I introduce everyone to the rich topic of concentration inequalities – inequalities that quantify how a random variable deviates around its mean. In particular, I will prove some basic inequalities: Hoeffding, Chernoff, and Bernstein. Then I will present two important classes of distributions: sub-gaussian and sub-exponential. These classes form a natural environment in which many results of high-dimensional probability and its applications are developed from.
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