A strong composition theorem for junta complexity and the boosting of property testers

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 $\varepsilon^{\prime}$-approximate junta complexity, even for $\varepsilon^{\prime} \gg \varepsilon$. 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.