The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform<\/i> distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions.<\/p>\n
In this paper, we give simple and optimal testers for bounded derivative properties<\/i> over arbitrary product distributions.<\/i> Bounded derivative properties include fundamental properties such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing.<\/p>\n
We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem<\/i> for all bounded derivative properties, which relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best known result.<\/p>\n","protected":false},"excerpt":{"rendered":"
The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. 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