Given $v_1,\ldots, v_m\in\mathbb{C}^d$ with $\|v_i\|^2= \alpha$ for all $i\in[m]$ as input and suppose $\sum_{i=1}^m | \langle u, v_i \rangle |^2 = 1$ for every unit vector $u\in\mathbb{C}^d$, Weaver's discrepancy problem asks for a partition $S_1, S_2$ of $[m]$, such that $\sum_{i\in S_{j}} |\langle u, v_i \rangle|^2 \leq 1 -\theta$ for some universal constant $\theta$, every unit vector $u\in\mathbb{C}^d$ and every $j\in\{1,2\}$. We prove that this problem can be solved deterministically in polynomial time when $m\geq 49 d^2$.

翻译：给定 $v_1,\ldots, v_m\in\mathbb{C}^d$，且对任意 $i\in[m]$ 满足 $\|v_i\|^2= \alpha$ 作为输入，假设对于每个单位向量 $u\in\mathbb{C}^d$ 有 $\sum_{i=1}^m | \langle u, v_i \rangle |^2 = 1$，韦弗偏差问题要求将 $[m]$ 划分为两个子集 $S_1, S_2$，使得对于某个普适常数 $\theta$，每个单位向量 $u\in\mathbb{C}^d$ 及每个 $j\in\{1,2\}$ 均有 $\sum_{i\in S_{j}} |\langle u, v_i \rangle|^2 \leq 1 -\theta$。我们证明当 $m\geq 49 d^2$ 时，该问题可在多项式时间内确定性求解。