Testing Non-uniform k-Wise Independent Distributions over Product Spaces

A discrete distribution D over Σ1 × · · · × Σn is called (non-uniform) k-wise independent if for any set of k indexes {i1, . . . , ik} and for any z1 ∈ Σi1 , . . . , zk ∈ Σik , PrX∼D[Xi1 · · · Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · · PrX∼D[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 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 when the underlying domain is {0, 1}n . 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 based on our results for the uniform case. These greatly generalize the results of Alon et al. [1] on uniform k-wise independence over the Boolean cubes to non-uniform k-wise inde...

• Type: Conference
• Year: 2010
• Where: ICALP
• Authors: Ronitt Rubinfeld, Ning Xie
• Keywords: ICALP 2010, K-wise Independence, K-wise Independent Distributions, Programming Languages, Uniform Case