Approximation of the average of some random matrices

Rudelson's theorem states that if for a set of unit vectors $u_i$ and positive weights $c_i$, we have that $\sum c_i u_i\otimes u_i$ is the identity operator $I$ on ${\mathbb R}^d$, then the sum of a random sample of $Cd\ln d$ 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 $d^2$ to approximate the identity.

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