Greedy sparsifications of sums of positive semidefinite matrices

Abstract: We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\dots,A_m$ be positive semidefinite (d\times d) matrices, and let $λ1,\dots,λ_m \ge 0$ satisfy [ \sum{i=1}^m λi = 1, \qquad \sum{i=1}^m λi A_i = I_d, \qquad |A_i| \le M \quad\text{for all } i=1,\dots,m. ] We show that there exists a deterministic sequence of indices $i_1,i_2,\dots \in {1,\dots,m}$ such that for every integer $k \ge 1$, [ \left| \frac{1}{k}\sum{r=1}^k A_{i_r} - I_d \right| \le \begin{cases} \displaystyle \frac{2M\ln(2d)}{k}, & \text{if } k \le M\ln(2d),\[2ex] \displaystyle 3\sqrt{\frac{M\ln(2d)}{k}}, & \text{if } k > M\ln(2d). \end{cases} ] In particular, if $0<\varepsilon\le 1$ and $N \ge 9M\ln(2d)\varepsilon^{-2}$, then one can choose indices $i_1,\dots,i_N \in {1,\dots,m}$ such that [ \left| \frac{1}{N}\sum_{r=1}^N A_{i_r} - I_d \right| \le \varepsilon. ]