Smoothed Analysis of the Komlós Conjecture: Rademacher Noise

07/12/2023

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by   Elad Aigner-Horev, et al.

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The discrepancy of a matrix M ∈ℝ^d × n is given by DISC(M) := min_x∈{-1,1}^nMx_∞. An outstanding conjecture, attributed to Komlós, stipulates that DISC(M) = O(1), whenever M is a Komlós matrix, that is, whenever every column of M lies within the unit sphere. Our main result asserts that DISC(M + R/√(d)) = O(d^-1/2) holds asymptotically almost surely, whenever M ∈ℝ^d × n is Komlós, R ∈ℝ^d × n is a Rademacher random matrix, d = ω(1), and n = ω̃(d^5/4). We conjecture that n = ω(d log d) suffices for the same assertion to hold. The factor d^-1/2 normalising R is essentially best possible.