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Revision #2 to TR13-124 | 12th March 2015 15:55

The Complexity of Deciding Statistical Properties of Samplable Distributions

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Revision #2
Authors: Thomas Watson
Accepted on: 12th March 2015 15:56
Downloads: 286
Keywords: 


Abstract:

We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies 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.


Revision #1 to TR13-124 | 16th September 2013 22:06

The Complexity of Deciding Statistical Properties of Samplable Distributions





Revision #1
Authors: Thomas Watson
Accepted on: 16th September 2013 22:06
Downloads: 728
Keywords: 


Abstract:

We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies 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.


Paper:

TR13-124 | 9th September 2013 04:33

The Complexity of Deciding Statistical Properties of Samplable Distributions





TR13-124
Authors: Thomas Watson
Publication: 9th September 2013 14:44
Downloads: 796
Keywords: 


Abstract:

We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies 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 introduce a bounded-error version of C=P, which we call BC=P, and we investigate its structural properties.



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