Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Abstract

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., $\xi_{i}\sim \mathcal{N}(0,1)$). We are interested to know under what conditions “typical norm” of the random matrix $S_N = \sum_{i=1}^N\xi_{i}B_{i}$ is of order of 1. An evident necessary condition is ${\bf E}\{S_{N}^{2}\}\preceq O(1)I$, which, essentially, translates to $\sum_{i=1}^{N}B_{i}^{2}\preceq I$; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is $\leq O(1)m^{{1\over 6}}$: ${\rm Prob}\{||S_N||>\Omega m^{1/6}\}\leq O(1)\exp\{-O(1)\Omega^2\}$ 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 ${\rm Opt} = \max\limits_{X_{j}\in{\bf R}^{m\times m}}\left\{F(X_1,\ldots ,X_k): X_jX_j^{\rm T}=I,\,j=1,\ldots ,k\right\}$, where F is quadratic in X =  (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices $X_j : {\rm Opt}\leq {\rm Opt}(SDP)\leq O(1) [m^{1/3}+\ln k]{\rm Opt}$.