# CSAIL Event Calendar: Previous Series

 Testing k-wise and Almost k-wise IndependenceSpeaker: Ning Xie , MITDate: December 11 2006 Time: 4:00PM to 5:15PM Location: 32-G575Host: Madhu Sudan, MIT CSAILContact: Joanne Hanley, 617 253 6054, joanne@csail.mit.eduRelevant URL: http://theory.csail.mit.edu/~madhu/algcomp/ning-abs.htmlIn this work, we consider the problems of testing whether a distribution over $\{0,1\}^n$ is $k$-wise or $(\eps,k)$-wise independent using samples drawn from that distribution. To distinguish $k$-wise independent distributions from those that are $\delta$-far in statistical distance, we upper bound the number of required samples by $\tilde{O}(n^k/\delta^{2})$ and lower bound it by $\Omega(n^{\frac{k-1}{2}}/\delta)$ (these bounds hold for constant $k$, and essentially the same bounds hold for general $k$). To achieve these bounds, we use Fourier analysis to relate a distribution's distance from $k$-wise independence to its \emph{biases} (a measure of the parity imbalance it induces on a set of variables). The relationships we derive are tighter than previously known, and are of independent interest. To distinguish $(\eps,k)$-wise independent distributions from those that are $\delta$-far in statistical distance, we upper bound the number of required samples by $O\left({k \log n \over \delta2\epsilon2}\right)$ and lower bound it by $\Omega\left({\sqrt{k\log n} \over (\eps+\delta)\sqrt{\log 1/(\eps+\delta)}}\right)$. Although these bounds are an exponential improvement (in terms of $n$ and $k$) over the corresponding bounds for testing $k$-wise independence, we show that the {\em time} complexity of testing $(\eps,k)$-wise independence is unlikely to be $\poly(n,1/\eps,1/\delta)$ for $k=\Theta(\log n)$, since this would disprove a plausible conjecture about the hardness of finding hidden cliques in random graphs. Under the conjecture, our result implies that for, say, $k =\log n$ and $\eps = 1 / n^{0.99}$, there is a set of $(\eps,k)$-wise independent distributions, and a set of distributions at distance $\delta=1/n^{0.51}$ from $(\eps,k)$-wise independence, which are indistinguishable by polynomial time algorithms. Joint work with Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, and Ronitt Rubinfeld.See other events that are part of Algorithms and Complexity Series 2006/2007See other events happening in December 2006