Approximation of the average of some random matrices

Abstract

Rudelson's theorem states that if for a set of unit vectors ui and positive weights ci, we have that ∑ciui⊗ui is the identity operator I on Rd, then the sum of a random sample of Cdln⁡d of these diadic products is close to I. The ln⁡d term cannot be removed.On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the ln⁡d term can be removed, if one wants to show the existence of a good approximation of I as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semi-definite matrices.First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by Cd elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices.Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach–Mazur distance of convex bodies.Finally, we show that in some cases, one needs to take a subset of the vectors of order d2 to approximate the identity.