Fast Spectral Signing for Vector Balancing

2026-09-24

AI Summary

This study addresses the high time complexity bottleneck encountered by algorithms that determine signs in vector balancing to bound the coordinates of matrix products. To overcome this, we propose a deterministic fast spectral signing algorithm that optimizes discretization via energy-corrected barriers and eigenvalue monitoring. Furthermore, it introduces moving clocks and weak quadratic charge mechanisms to limit update counts, while integrating polynomial soft projectors, affine row representation repairs, and matrix multiplication to handle constraint residuals. This work reduces the number of updates to O(n²log n), substantially decreases the discrepancy constant from 8272 to 99, and improves the running time to ~O(mn+n^4.372). Ultimately, the proposed approach achieves near-linear logarithmic acceleration while maintaining polynomial bit complexity.

Abstract

The Komlós conjecture, now a theorem, asserts that whenever the columns of a matrix $A\in\mathbb{R}^{m\times n}$ have Euclidean norm at most one, some signs $\varepsilon\in\{-1,1\}^n$ make every coordinate of $A\varepsilon$ bounded by an absolute constant. Guo, Fang, and Lu gave the first polynomial-time algorithm for finding such signs, a deterministic spectral signing procedure with discrepancy $8272$ and running time $O((mn^9+n^{10})\log(m+n))$. We give a deterministic algorithm that finds signs with $\|A\varepsilon\|_\infty<99$ using $O(mn+n^{ω+2}\log^3 n)$ arithmetic operations, where $ω>2$ is any fixed attainable matrix-multiplication exponent; with the current bounds on $ω$ this is $\widetilde O(mn+n^{4.372})$. Our algorithm uses the same framework: it rounds a single fractional coloring and watches all rows through the top eigenvalue of a Gram matrix of energy-corrected barriers. Steps follow flat directions, rescaled so that no barrier near its threshold moves faster than a constant, and the regularizer grows as coordinates freeze; together these bound the number of updates by $O(n^2\log n)$. A weak quadratic charge on the tracked row sums leaves $O(n\log^2 n)$ rows to evaluate at any time, and a motion clock bounds when any other row could approach its barrier. Each update is a short sequence of matrix products. Its direction is read off by conditional expectations from a polynomial soft projector, its small constraint residual is repaired in affine row representations, and one identity accounts for every change of representation. For rational input the algorithm has polynomial bit complexity.

Problem

• Komlós conjecture
• vector balancing
• spectral signing
• discrepancy

Innovation

• Spectral Signing
• Vector Balancing
• Komlós Conjecture
• Discrepancy Minimization
• Barrier Methods