Information theory in property testing and monotonicity testing in higher dimension

In property testing, we are given oracle access to a function f, and we wish to test if the function satisfies a given property P, or it is ϵ-far from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the domain. This paper relates the complexity of testing the monotonicity of a function over the d-dimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the query complexity of the property tester.