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

A distribution D over Σ1 × ... × Σn is called (non-uniform) k-wise independent if for any set of k indices {i1,..., ik} and for any z1...zk ∈ Σi1×...×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 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.