Testing monotone high-dimensional distributions

Ronitt Rubinfeld, Rocco A. Servedio

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


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 ε-far (in the L 1 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 ε-far from a fixed known monotone product distribution and approximating the entropy of an unknown monotone distribution).

Original languageEnglish
Pages (from-to)24-44
Number of pages21
JournalRandom Structures and Algorithms
Issue number1
StatePublished - Jan 2009
Externally publishedYes


  • Distribution testing
  • Monotone distributions
  • Property testing
  • Sublinear algorithms


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