Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the \emph{tolerant testing} of juntas. Given black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$, we give a $poly(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k'$-juntas, where $k' = O(\frac{k}{\varepsilon^2})$. In the non-relaxed setting, we extend our ideas to give a $2^{\tilde{O}(\sqrt{k/\varepsilon})}$ (adaptive) query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k$-juntas. To the best of our knowledge, this is the first subexponential-in-$k$ query algorithm for approximating the distance of $f$ to being a $k$-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in $k$). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [AoP, 1994].

中文翻译：

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使用次指数查询的军塔距离近似

利用 De、Mossel 和 Neeman [FOCS，2019] 的工具，我们展示了两种不同的结果，这些结果与军政府的 \emph {宽容测试}有关。给定对布尔函数 $f:\{\pm1\}^{n} \to \{\pm1\}$ 的黑盒访问，我们给出一个 $poly(k, \frac{1}{\varepsilon}) $ 查询算法区分$\gamma$-接近$k$-juntas 和$(\gamma+\varepsilon)$-远离$k'$-juntas 的函数，其中$k' = O(\frac{ k}{\varepsilon^2})$。在非松弛设置中，我们扩展我们的想法，给出一个 $2^{\tilde{O}(\sqrt{k/\varepsilon})}$（自适应）查询算法，它区分了 $\gamma$-接近 $k$-juntas 和 $(\gamma+\varepsilon)$-远离 $k$-juntas。据我们所知，这是第一个 subexponential-in-$k$ 查询算法，用于近似 $f$ 到 $k$-junta 的距离（Blais、Canonne、Eden、Levi 和 Ron [SODA，2018] 和 De 、Mossel 和 Neeman [FOCS, 2019] 需要以 $k$ 为单位的指数级查询）。我们的技术是傅立叶分析的，并利用了 Talagrand [AoP, 1994] 引入的“标准化影响”的概念。