We prove a strong composition theorem for junta complexity and show how such theorems can be used to generically boost the performance of property testers. The $\varepsilon$-approximate junta complexity of a function $f$ is the smallest integer $r$ such that $f$ is $\varepsilon$-close to a function that depends only on $r$ variables. A strong composition theorem states that if $f$ has large $\varepsilon$-approximate junta complexity, then $g \circ f$ has even larger $\varepsilon'$-approximate junta complexity, even for $\varepsilon' \gg \varepsilon$. We develop a fairly complete understanding of this behavior, proving that the junta complexity of $g \circ f$ is characterized by that of $f$ along with the multivariate noise sensitivity of $g$. For the important case of symmetric functions $g$, we relate their multivariate noise sensitivity to the simpler and well-studied case of univariate noise sensitivity. We then show how strong composition theorems yield boosting algorithms for property testers: with a strong composition theorem for any class of functions, a large-distance tester for that class is immediately upgraded into one for small distances. Combining our contributions yields a booster for junta testers, and with it new implications for junta testing. This is the first boosting-type result in property testing, and we hope that the connection to composition theorems adds compelling motivation to the study of both topics.

翻译：我们证明了关于集合复杂度的一个强组合定理，并展示了此类定理如何用于通用的性质测试器性能提升。一个函数 $f$ 的 $\varepsilon$ -近似集合复杂度是指使得 $f$ 与某个仅依赖于 $r$ 个变量的函数在 $\varepsilon$ 误差内接近的最小整数 $r$。强组合定理指出：若 $f$ 具有较大的 $\varepsilon$ -近似集合复杂度，则 $g \circ f$ 将具有更大的 $\varepsilon'$ -近似集合复杂度，即使 $\varepsilon' \gg \varepsilon$ 时也成立。我们对此行为建立了较为完整的理解，证明 $g \circ f$ 的集合复杂度由 $f$ 的集合复杂度以及 $g$ 的多变量噪声敏感度共同刻画。对于对称函数 $g$ 这一重要情形，我们将其多变量噪声敏感度关联到更简单且已被充分研究的单变量噪声敏感度。进一步，我们展示了强组合定理如何为性质测试器提供提升算法：对于任意函数类，若存在该类的强组合定理，则可立即将针对该类的远距离测试器升级为近距离测试器。综合我们的贡献，我们得到了集合测试器的提升器，并由此得出关于集合测试的新推论。这是性质测试领域首个提升类结果，我们期望与组合定理的关联能为这两个方向的研究增添强有力的动机。