Polynomial-Time Algorithms for Weaver's Discrepancy Problem in a Dense Regime

Abstract: 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 $m\geq 49 d^2$.