Testing monotone high-dimensional distributions

A <i>monotone distribution</i> <i>P</i> over a (partially) ordered domain has <i>P</i>(<i>y</i>) ≥ <i>P</i>(<i>x</i>) if <i>y</i> ≥ <i>x</i> in the order. We study several natural problems of testing properties of monotone distributions over the <i>n</i>-dimensional Boolean cube, given access to random draws from the distribution being tested. We give a poly(<i>n</i>)-time algorithm for testing whether a monotone distribution is equivalent to or ε-far (in the <i>L</i><inf>1</inf> 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 <i>n</i>-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(<i>n</i>) 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).