Four Deviations Suffice for Rank 1 Matrices

Abstract: We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables $\xi_1, \ldots, \xi_n$ with finite support, e.g. ${ \pm 1 }$ or ${ 0,1 }$-valued random variables, or some combination thereof. Let $u_1, \dots, u_n \in \mathbb{C}^m$ and $$ \sigma^2 = \left| \sum_{i=1}^n \text{Var} \xi_i ^2 \right|. $$ Then there exists a choice of outcomes $\varepsilon_1,\ldots,\varepsilon_n$ in the support of $\xi_1, \ldots, \xi_n$ s.t. $$ \left |\sum_{i=1}^n \mathbb{E} [ \xi_i] u_i u_i^* - \sum_{i=1}^n \varepsilon_i u_i u_i^* \right | \leq 4 \sigma. $$ A simple consequence of our result is an improvement of a Lyapunov-type theorem of Akemann and Weaver.