Smoothed Analysis of the Komlós Conjecture
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The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the ℓ_∞ 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 =ω(dlog 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.
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