Testing monotone high-dimensional distributions

A monotone distribution P over a (partially) ordered domain assigns higher probability to y than to x if y x in the order. We study several natural problems concerning 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 -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 various problems of 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, including testing whet...

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Ronitt Rubinfeld, Rocco A. Servedio
• Keywords: Discrete Algorithms, Monotone Distribution, SODA 2010, Testing Monotone Distributions, Unknown Monotone Distribution