Galyna Livshyts - A Tight Net with Respect to a Random Matrix Norm and Applications to Estimating Singular Values

In this talk we construct a net around the unit sphere with good properties. We show that with exponentially high probability, the value of |Ax| on the sphere can be approximated well using this net, where A is any random matrix with independent columns. We apply it to study the smallest singular value of random matrices under very mild assumptions, and obtain sharp small ball behavior.