Continuous Optimization Seminar - Courtney Paquette

Title: High-dimensional probability: Random vectors in high dimensions

Abstract:

In this talk, I will finish our discussion of concentration inequalities, particularly, I will discuss the sub-exponential distribution and state Bernstein’s inequality; thereby completing our study of large deviations. Following this, I will introduce random vectors which live in high-dimensions. Life in high dimensions presents new challenges which stems from the fact that there is exponentially more room in high dimensions than in lower dimensions. The abundance of room in higher dimensions makes algorithmic tasks exponentially more difficult. Probability in high dimensions will give us some tools to circumvent these difficulties. I will start by examining the Euclidean norm of a random vector X with independent coordinates, and show that the norm concentrates tightly about its mean. Then I will state and prove some basic results (e.g. isotropic distributions) and give examples of high-dimensional distributions (e.g. multivariant normal, spherical, etc). Finally, I will examine principle component analysis (PCA), a tool which is of utmost importance in probability, statistics, data science, and optimization.