Lecture 2: Kolmogorov and Gelfand widths and Kashin’s inequalities
Feng Dai教授系列讲座
报告题目(Title)：Lecture 2: Kolmogorov and Gelfand widths and Kashin’s inequalities
报告人(Speaker)：Feng Dai 教授 (University of Alberta, Canada)
地点(Place)：后主楼1220室
时间(Time)：2019年12月21日上午8:00至12:00
邀请人(Inviter)：杨大春
报告摘要
Gelfand and Kolmogorov widths are important concepts in optimality and high-dimensional approximation. For instance, they have interesting applications in the fields of compressive sensing and sparse approximation because they give general performance bounds for sparse recovery methods. In this lecture, I will first review some basic properties of the Kolmogorov and Gelfand widths. I will focus on the basic idea behind the definition, and explain why these concepts are important. After that, I will show how the concentration inequalities in probability can be used to prove the celebrated Kashin inequality on the Kolmogorov $n$-widths of the unit ball of $\mathbb{R}^m$ in the metric $\ell_\infty^m$, where $m$ is significantly larger than $n$.
