AMS Without 4-Wise Independence on Product Domains

TL;DR

This paper extends sketching techniques based on 4-wise independence to product domains, enabling efficient one-pass approximation of joint distributions and independence testing for high-dimensional data streams.

Contribution

It generalizes existing methods from pairs to k-ary vectors, providing a space-efficient randomized algorithm for independence approximation in data streams.

Findings

Achieves a $(1 \u00b1 \u03b5)$ approximation with high probability.

Uses space logarithmic in domain sizes and exponential in dimension.

Extends previous work from 2-dimensional to k-dimensional data.

Abstract

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.