Testing non-uniform k-wise independent distributions over product spaces (extended abstract)
Author(s)
Rubinfeld, Ronitt; Xie, Ning
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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.
Date issued
2010-07Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer ScienceJournal
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.
Version: Author's final manuscript
ISSN
0170-1495