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

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 $z_{1}\cdots z_{k} \in \Sigma_{i_{1}}\times \cdots \times \Sigma_{i_{k}}$, $\Pr_{X\sim D}[X_{i_{1}}\cdots X_{i_{k}}\!=z_{1}\cdots z_{k}] =\Pr_{X\sim D}[X_{i_{1}}=z_{1}] \cdots \Pr_{X\sim D}[X_{i_{k}}=z_{k}]$. 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.

A full version of this paper is available at http://people.csail.mit.edu/ningxie/papers/RX09.pdf