哪些分布距离是次线性可测的？
Which Distribution Distances are Sublinearly Testable?
摘要：
给定来自未知分布$p$的样本和分布$q$的描述，$p$和$q$是接近还是远离？在测试$p$和$q$在总变异距离上是否相等或远的情况下，“同一性测试”的问题已经受到了显著的关注。然而，在最近的工作中，以下问题对于解决分布测试前沿的问题至关重要：-替代距离：我们可以测试$p$和$q$是否在其他距离上很远，Hellinger说？- 容差：当$p$和$q$接近而不是相等时，我们可以测试吗？如果是的话，距离有多远？基于这些问题，我们描述了各种距离下分布检验的复杂性，包括全变差、$\ell_2$、Hellinger、Kullback-Leibler和$χ^2$。对于每对距离$d_1$和$d_2$，我们研究了如果$p$和$q$在$d_1$中接近而在$d_2$中远离，则测试的复杂性，重点在于识别哪些问题允许强次线性测试器（即，对于某些$γ> 0$具有复杂度$O(n^{1 - γ})$的那些，其中$n$是分布$p$和$q$的支持的大小）。我们为每种情况提供匹配的上限和下限。我们还在只有$q$样本（等效性试验）的情况下研究了这些问题，在何时可以达到耐受性方面与同一性试验存在定性差异。我们的算法属于$χ^2$ -统计的经典范例，但需要关键的变化来处理我们考虑的每个距离所带来的挑战。最后，我们调查其他最近的结果，试图作为一个参考的各种分布测试问题的复杂性。
英文摘要：
Given samples from an unknown distribution $p$ and a description of a distribution $q$, are $p$ and $q$ close or far? This question of "identity testing" has received significant attention in the case of testing whether $p$ and $q$ are equal or far in total variation distance. However, in recent work, the following questions have been been critical to solving problems at the frontiers of distribution testing: -Alternative Distances: Can we test whether $p$ and $q$ are far in other distances, say Hellinger? -Tolerance: Can we test when $p$ and $q$ are close, rather than equal? And if so, close in which distances? Motivated by these questions, we characterize the complexity of distribution testing under a variety of distances, including total variation, $\ell_2$, Hellinger, Kullback-Leibler, and $χ^2$. For each pair of distances $d_1$ and $d_2$, we study the complexity of testing if $p$ and $q$ are close in $d_1$ versus far in $d_2$, with a focus on identifying which problems allow strongly sublinear testers (i.e., those with complexity $O(n^{1 - γ})$ for some $γ> 0$ where $n$ is the size of the support of the distributions $p$ and $q$). We provide matching upper and lower bounds for each case. We also study these questions in the case where we only have samples from $q$ (equivalence testing), showing qualitative differences from identity testing in terms of when tolerance can be achieved. Our algorithms fall into the classical paradigm of $χ^2$-statistics, but require crucial changes to handle the challenges introduced by each distance we consider. Finally, we survey other recent results in an attempt to serve as a reference for the complexity of various distribution testing problems.
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