Abstract
We consider the problems of deciding whether the joint distribution sampled by a given circuit has certain statistical properties such as being i. i. d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all these problems are C=P-complete, by showing that the following promise problem (which is a restriction of all the above problems) is C=P-complete: Given a circuit, distinguish the case where the output distribution is uniform and the case where every pair of coordinates is neither uncorrelated nor identically distributed. This completeness result holds even for samplers that are depth-3 circuits. We also consider circuits that are d-local, in the sense that each output bit depends on at most d input bits. We give linear-time algorithms for deciding whether a 2-local sampler’s joint distribution is fully independent, and whether it is exchangeable. We also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZK-complete. We also introduce a bounded-error version of C=P, which we call BC=P, and we investigate its structural properties. ACM Classification: F.1.3 AMS Classification: 68Q17, 68Q15
Highlights
Testing for independence of random variables is a fundamental problem in statistics
One of the most general and natural ways to succinctly specify a distribution is to give the code of an efficient algorithm that takes “pure” randomness and transforms it into a sample from the distribution. (This gives a polynomial-size specification of a distribution over a potentially exponential-size set.) For arbitrary circuit samplers, the papers [31, 20, 21, 40] contain completeness results for various approximation problems concerning statistical distance, Shannon entropy, and min-entropy
We prove that our C=P-completeness results hold even when restricted to samplers that are AC0-type circuits with depth 3 and top fan-in 2 (i. e., each output gate has fan-in at most 2)
Summary
Testing for independence of random variables is a fundamental problem in statistics. Theoretical computer scientists have studied this and other analogous problems from two main viewpoints. Papers that study AC0 circuits that sample distributions include [36, 27, 37, 8] Another (generally more restrictive) model of efficient parallel time computation is locally-computable functions, where each output bit depends on at most a bounded number of input bits. We consider approximate versions of the problems discussed above: deciding whether the joint distribution of a given sampler is statistically close to or far from satisfying a property. It was shown in [20] that for the property of being uniform, the problem is complete for the class NISZK (non-interactive statistical zero-knowledge). It does not appear to be directly relevant to statistical properties of samplable distributions, we take the opportunity to study this class and prove that it is closed under several operations (disjunction, conjunction, union, and intersection)
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