关于k-均匀性的接近性

摘要：

{-1，1}^n上的概率分布是（eps，k）-均匀的，如果粗略地说，当限制到任何k坐标时，它是eps-接近均匀分布。我们考虑的问题有多远（eps，k）-明智的均匀分布可以从任何全球k-明智的均匀分布。我们证明了每个（eps，k）均匀分布在总变差距离上都是O（n^（k/2）eps）-接近于k均匀分布。此外，我们还证明了该界对于所有偶数k都是最优的：我们发现了一个（eps，k）方向的均匀分布，即Omega（n^（k/2）eps）-在总变差距离中远离任何k方向的均匀分布。对于k = 1，我们得到了O（eps）的更好的上界，这也是最优的。我们的贴近度结果的一个应用是测试分布是否是k-wise均匀或远离k-wise均匀的样本复杂度。我们在所需的样本上给出了O（n^k/delta^2）（或当k = 1时O（log n/delta^2））的上界。对于完全一致检验和远离k-一致检验的特殊情况，我们给出了O~（n^（k/2）/delta^2）的改进的上界.最后，当k = 2时，我们用匹配的Ω（n/delta^2）下界来补充这一点。我们的结果改进了[AAK+07]中的最佳界，并且具有更简单的证明。

英文摘要：

A probability distribution over {-1, 1}^n is (eps, k)-wise uniform if, roughly, it is eps-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (eps, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (eps, k)-wise uniform distribution is O(n^(k/2) eps)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (eps, k)-wise uniform distribution that is Omega(n^(k/2) eps)-far from any k-wise uniform distribution in total variation distance. For k = 1, we get a better upper bound of O(eps), which is also optimal. One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^k/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^(k/2)/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2. Our results improve upon the best known bounds from [AAK+07], and have simpler proofs.

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