有限域上函数的凸性检验
Testing convexity of functions over finite domains
摘要：
我们建立了新的上限和下限的数量所需的查询，以测试凸函数在各种离散域。1.我们提供了一个简化版本的非自适应凸度测试仪在线。我们重新证明了通常一致模型下的上界$O(\frac{\log(εn)}ε)$，并证明了分布自由情形下的上界$O(\frac{\log n}ε)$. 2.我们展示了一个紧密的下界$Ω(\frac{\log(εn)}ε)$查询测试凸性的功能$f: [n] \rightarrow \mathbb{R}$上的线。该下限适用于自适应和非自适应算法，并与第1项的上限相匹配，表明自适应在此设置中没有帮助。3.转向更高维度，我们考虑条带$[3] \times [n]$的情况。我们构造了一个\emph{adaptive}函数凸性测试器$f\colon [3] \times [n] \to \mathbb R$，具有查询复杂度$O(\log^2 n)$。我们还表明，任何\emph{non-adaptive}测试人员必须在此设置中使用$Ω(\sqrt{n})$查询。因此，自适应产生指数改善这个问题。4.对于功能$f\colon [n]^d \to \mathbb R$域的尺寸$d \geq 2$，我们显示了一个非自适应查询下限$Ω((\frac{n}{d})^{\frac{d}{2}})$。
英文摘要：
We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the upper bound $O(\frac{\log(εn)}ε)$ in the usual uniform model, and prove an $O(\frac{\log n}ε)$ upper bound in the distribution-free setting. 2. We show a tight lower bound of $Ω(\frac{\log(εn)}ε)$ queries for testing convexity of functions $f: [n] \rightarrow \mathbb{R}$ on the line. This lower bound applies to both adaptive and non-adaptive algorithms, and matches the upper bound from item 1, showing that adaptivity does not help in this setting. 3. Moving to higher dimensions, we consider the case of a stripe $[3] \times [n]$. We construct an \emph{adaptive} tester for convexity of functions $f\colon [3] \times [n] \to \mathbb R$ with query complexity $O(\log^2 n)$. We also show that any \emph{non-adaptive} tester must use $Ω(\sqrt{n})$ queries in this setting. Thus, adaptivity yields an exponential improvement for this problem. 4. For functions $f\colon [n]^d \to \mathbb R$ over domains of dimension $d \geq 2$, we show a non-adaptive query lower bound $Ω((\frac{n}{d})^{\frac{d}{2}})$.
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