Testing monotone high-dimensional distributions

• Authors: Ronitt Rubinfeld;Rocco A. Servedio
• Venue: Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
• Year: 2009

A monotone distribution P over a (partially) ordereddomain has P(y) ≥ P(x) if y≥ x in the order. We study several natural problems oftesting properties of monotone distributions over then-dimensional Boolean cube, given access to random drawsfrom the distribution being tested. We give a poly(n)-timealgorithm for testing whether a monotone distribution is equivalentto or ε-far (in the L1 norm) fromthe uniform distribution. A key ingredient of the algorithm is ageneralization of a known isoperimetric inequality for the Booleancube. We also introduce a method for proving lower bounds ontesting monotone distributions over the n-dimensionalBoolean cube, based on a new decomposition technique for monotonedistributions. We use this method to show that our uniformitytesting algorithm is optimal up to polylog(n) factors, andalso to give exponential lower bounds on the complexity of severalother problems (testing whether a monotone distribution isidentical to or ε-far from a fixed known monotoneproduct distribution and approximating the entropy of an unknownmonotone distribution). © 2008 Wiley Periodicals, Inc. RandomStruct. Alg., 2009A preliminary version of this work appeared in the 2005 ACMSymposium on Theory of Computing (STOC) (see 12).Any opinions, findings, and conclusions or recommendationsexpressed in this article are those of the authors and do notnecessarily reflect the views of the National ScienceFoundation.