Testing monotone high-dimensional distributions

A monotone distribution P over a (partially) ordered domain has P(y) ≥ P(x) if y ≥ x in the order. We study several natural problems of testing properties of monotone distributions over the n-dimensional Boolean cube, given access to random draws from the distribution being tested. We give a poly(n)-time algorithm for testing whether a monotone distribution is equivalent to or e-far (in the L1 norm) from the uniform distribution. A key ingredient of the algorithm is a generalization of a known isoperimetric inequality for the Boolean cube. We also introduce a method for proving lower bounds on testing monotone distributions over the n-dimensional Boolean cube, based on a new decomposition technique for monotone distributions. We use this method to show that our uniformity testing algorithm is optimal up to polylog(n) factors, and also to give exponential lower bounds on the complexity of several other problems (testing whether a monotone distribution is identical to or e-far from a fixed known monotone product distribution and approximating the entropy of an unknown monotone distribution). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 A preliminary version of this work appeared in the 2005 ACM Symposium on Theory of Computing (STOC) (see 12). Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Science Foundation.