Sums of Random Symmetric Matrices and Applications

Let Bi be deterministic symmetric m ×m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested in conditions for the “typical norm” of the random matrix SN = N ∑ i=1 ξiBi to be of order of 1. An evident necessary condition is E{S2 N} 1 O(1)I, which, essentially, translates to N ∑ i=1 B i 1 I; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, show that under the above condition the typical norm of SN is ≤ O(1)m 6 : Prob{‖SN‖ > Ωm1/6} ≤ O(1) exp{−O(1)Ω2} for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints Opt = max Xj∈Rm×m { F (X1, ..., Xk) : XjX j = I, j = 1, ..., k } , where F is quadratic in X = (X1, ..., Xk). We show that when F is convex in every one of Xj , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices Xj : Opt ≤ Opt(SDP ) ≤ O(1)[m + ln k]Opt. AMS Subject Classification: 60F10, 90C22, 90C25, 90C59.