Smoothed Analysis of the Koml\'os Conjecture

Abstract

The well-known Koml\'os conjecture states that given $n$ vectors in $\mathbb{R}^d$ with Euclidean norm at most one, there always exists a $\pm 1$ coloring such that the $\ell_{\infty}$ norm of the signed-sum vector is a constant independent of $n$ and $d$. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors $n =\omega(d\log d)$. The dependence of $n$ on $d$ is the best possible even in a completely random setting. Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.