关于Junta复杂度的强合成定理与性质测试器的提升
A Strong Composition Theorem for Junta Complexity and the Boosting of Property Testers
AI总结：
本文证明了junta复杂度的强合成定理，揭示了复合函数junta复杂度与原函数复杂度及外层函数噪声敏感度的关系，并据此首次在性质测试中实现了测试器的通用提升。
摘要：
我们证明了一个关于junta复杂度的强合成定理，并展示了如何利用这类定理来通用地提升性质测试器的性能。函数$f$的$\varepsilon$-近似junta复杂度是指最小的整数$r$，使得$f$与某个仅依赖于$r$个变量的函数$\varepsilon$-接近。强合成定理表明，如果$f$具有较大的$\varepsilon$-近似junta复杂度，那么$g \circ f$具有更大的$\varepsilon'$-近似junta复杂度，即使$\varepsilon' \gg \varepsilon$。我们对这一行为建立了相当完整的理解，证明了$g \circ f$的junta复杂度由$f$的junta复杂度以及$g$的多元噪声敏感度共同刻画。对于重要的对称函数$g$情形，我们将其多元噪声敏感度与更简单且已被充分研究的单变量噪声敏感度联系起来。随后我们展示了强合成定理如何为性质测试器提供提升算法：对于任意函数类，只要拥有该类的强合成定理，一个用于大距离的测试器就可以立即升级为适用于小距离的测试器。将我们的贡献结合起来，可以得到一个针对junta测试器的提升器，并由此为junta测试带来新的推论。这是性质测试领域第一个提升型结果，我们希望与合成定理的这一联系能为这两个方向的研究增添令人信服的动机。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
