On the discrepancy of jittered sampling Journal Article

Author(s): Pausinger, Florian; Steinerberger, Stefan
Article Title: On the discrepancy of jittered sampling
Affiliation IST Austria
Abstract: We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]d is generated for fixed m∈ℕ by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 1/10 d/N1/2+1/2d ≤EDN∗(P)≤ √d(log N) 1/2/N1/2+1/2d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points.
Keywords: Inverse of the star discrepancy; Jittered sampling; L2-discrepancy; Star discrepancy
Journal Title: Journal of Complexity
Volume: 33
ISSN: 1090-2708
Publisher: Academic Press  
Date Published: 2016-04-01
Start Page: 199
End Page: 216
DOI: 10.1016/j.jco.2015.11.003
Notes: We are grateful to the referee whose suggestions greatly improved the quality and clarity of the exposition.
Open access: yes (repository)
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