Measuring Independence of Datasets

Abstract: A data stream model represents setting where approximating pairwise, or $k$-wise, independence with sublinear memory is of considerable importance. In the streaming model the joint distribution is given by a stream of $k$-tuples, with the goal of testing correlations among the components measured over the entire stream. In the streaming model, Indyk and McGregor (SODA 08) recently gave exciting new results for measuring pairwise independence. The Indyk and McGregor methods provide $\log{n}$-approximation under statistical distance between the joint and product distributions in the streaming model. Indyk and McGregor leave, as their main open question, the problem of improving their $\log n$-approximation for the statistical distance metric. In this paper we solve the main open problem posed by of Indyk and McGregor for the statistical distance for pairwise independence and extend this result to any constant $k$. In particular, we present an algorithm that computes an $(\epsilon, \delta)$-approximation of the statistical distance between the joint and product distributions defined by a stream of $k$-tuples. Our algorithm requires $O(({1\over \epsilon}\log({nm\over \delta}))^{(30+k)^k})$ memory and a single pass over the data stream.