Testing juntas

We show that a boolean valued function over n variables, where each variable ranges in an arbitrary probability space, can be tested for the property of depending on only J of them using a number of queries that depends only polynomially on J and the approximation parameter ε. We present several tests that require a number of queries that is polynomial in J and linear in ε^-1. We show a non-adaptive test that has one-sided error, an adaptive version of it that requires fewer queries, and a non-adaptive two-sided version of the test that requires the least number of queries. We also show a two-sided non-adaptive test that applies to functions over n boolean variables, and has a more compact analysis. We then provide a lower bound of Ω̃(J) on the number of queries required for the non-adaptive testing of the above property; a lower bound of Ω(log(J+1)) for adaptive algorithms naturally follows from this. In establishing this lower bound we also prove a result about random walks on the group Z^q₂ that may be interesting in its own right. We show that for some t(q)=Õ(q ²), the distributions of the random walk at times t and t+2 are close to each other, independently of the step distribution of the walk. We also discuss related questions. In Particular, when given in advance a known J-junta function h, we show how to test a function for the property of being identical to h up to a permutation of the variables, in a number of queries that is polynomial in J and ε^-1.