Testing non-uniform k-wise independent distributions over product spaces (extended abstract)
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Author(s)
Rubinfeld, Ronitt
Xie, Ning
Date Issued
July 2010
Journal
Automata, Languages and Programming
Publisher
Springer-Verlag
Citation
Rubinfeld, Ronitt, and Ning Xie. “Testing Non-uniform k-Wise Independent Distributions over Product Spaces.” Automata, Languages and Programming. Ed. Samson Abramsky et al. Vol. 6198. Berlin: Springer Berlin Heidelberg, 2010. 565–581. (Lecture notes in computer science ; 6198) Web.
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Author's final manuscript
Abstract
A distribution D over Σ1× ⋯ ×Σ n is called (non-uniform) k-wise independent if for any set of k indices {i 1, ..., i k } and for any z1zki1ik, PrXD[Xi1Xik=z1zk]=PrXD[Xi1=z1]PrXD[Xik=zk]. We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al. [1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/978-3-642-14165-2_48
