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

TL;DR

Weaver's discrepancy (KS2) seeks a two-way partition of vectors in $\mathbb{C}^d$ with uniform squared norms so that every unit vector induces a uniformly small energy on each side. The paper presents two deterministic polynomial-time algorithms that solve KS2 in the dense regime, achieving a bound of $\le \frac{3}{4}$ on the per-vector energy for all unit vectors when $m$ is sufficiently large (specifically $m \ge 221 d^2$ for the first method and $m \ge 49 d^2$ for the second). The first algorithm relies on a logarithmic potential $\Phi^{u}(A) = -\log\det(uI - A)$ with a gradually increasing barrier and a determinant-maximizing vector selection; the second fixes the barrier and uses the Matrix Determinant Lemma to bound the determinant of $I - A_{m/2}$ via a sequence of rank-1 updates. Together, these results establish a non-random, polynomial-time solution to KS2 in the dense regime and illuminate a tight connection to determinant optimisation and spectral structure via barrier-based and MDL-based analyses.

Abstract

Given $v_1,\ldots, v_m\in\mathbb{C}^d$ with $\|v_i\|^2= α$ 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 $\sum_{i\in S_{j}} |\langle u, v_i \rangle|^2 \leq 1 -θ$ for some universal constant $θ$, 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 $m\geq 49 d^2$.

Theorems & Definitions (25)

• Theorem 1
• Remark 1
• Remark 2
• Lemma 2.1
• proof
• Lemma 2.2: Theorem 3.30, alma9924127819402466
• Lemma 2.3
• proof
• Lemma 2.4: Matrix Determinant Lemma, DING20071223
• Lemma 2.5