Measure Concentration and Non-asymptotic Singular Values Distributions of Random Matrices

Abstract

This thesis is devoted to the non-asymptotic random matrix theory and measure concentration phenomenon. We focus on using concentration inequalities together with other probabilistic and geometric methods to study singular values distributions of several types of random matrices. In Chapter II, we apply concentration inequalities to a convex geometry problem, namely upper bound for the Dvoretzky dimension in Milman-Schechtman theorem. Our approach combines properties of random projections and geometric observation. In Chapter III, we study the non-asymptotic distributions of all singular values for i.i.d. sub-gaussian matrices. We prove a non-asymptotic upper bound for all singular values of i.i.d. sub-gaussian matrices under some weak condition. It is the first tight non-asymptotic upper bound for all singular values other than Gaussian matrices. The upper bound provides a two-side bound together with known lower bound. In Chapter IV, we study the smallest singular values distributions of symmetric sparse matrices. We show that an n-dimensional sparse symmetric random matrix A is invertible with high probability under some condition on its sparsity level.