The Complexity of Deciding Statistical Properties of Samplable Distributions
TL;DR: It is shown that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZK-complete, and a bounded-error version of C=P is introduced, which is called BC=P, and its structural properties are investigated.
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
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TL;DR: This work proves nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over n elements, and they wish to estimate within ±ε the statistical distance between their distributions.
Abstract: We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over n elements, and they wish to estimate within ±e the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log n factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater Than is Θ (n log n).
13 citations
01 Jan 2014
TL;DR: A proof that simultaneously shows all the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties are C_{=P}-complete is given, by showing that the following promise problem is C-complete.
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.
2 citations
References
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20 Apr 2009
TL;DR: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.
Abstract: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.
2,965 citations
"The Complexity of Deciding Statisti..." refers background in this paper
...This could potentially make some problems easier, but there are complexity-theoretic results showing that several such problems are computationally hard, particularly when the input is a succinct description of a distribution....
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04 May 1997
TL;DR: A pseudo-random generator which produces n instances of a problem for which the analog of the XOR lemma holds is given, and it is shown that if any problem in E = DTIAl E(2°t”j) has circuit complexity 2Q(”), then P = BPP.
Abstract: Russell Impagliazzo* Avi Wigdersont Department of Computer Science Institute of Computer Science University of California Hebrew University San Diego, CA 91097-0114 Jerusalem, Israel russell@cs .ucsd. edu avi@cs .huj i. ac. il Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem becomes almost completely unpredictable. In this paper we show that, in non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random generator which produces n instances of a problem for which the analog of the XOR lemma holds. Combining this generator with the results of [25, 6] gives substantially improved results for hardness vs randomness tradeoffs. In particular, we show that if any problem in E = DTIAl E(2°t”j) has circuit complexity 2Q(”), then P = BPP. Our generator is a combination of two known ones the random walks on expander graphs of [1, 10, 19] and the nearly disjoint subsets generator of [23, 25]. The quality of the generator is proved via a new proof of the XOR lemma which may be useful for other direct product results. *Research supported by NSF YI Award CCR-92s70979, Sloan Research Fellowship BR-3311, grant #93025 of the joint US-Czechoslovak Science and Technology Program, and USA-Israel BSF Grant 92-00043 tWork pmtly done while visiting the Institute for Advanced Study, Princeton, N. J. 08540 and Princeton University. Research supported the Sloan Foundation, American-Israeli BSF grant 92-00106, and the Wolfson Research Awards, administered by the Israel Academy of Sciences.
692 citations
Book•
28 Apr 2008TL;DR: It is the view is that these (
on-technical") aspects are the core of the field, and the presentation attempts to re ect this view.
Abstract: This book is rooted in the thesis that complexity theory is extremely rich in conceptual content, and that this contents should be explicitly communicated in expositions and courses on the subject. The desire to provide a corresponding textbook is indeed the motivation for writing the current book and its main governing principle.The book offers a conceptual perspective on complexity theory, and the presentation is designed to highlight this perspective. It is intended mainly for students that wish to learn complexity theory and for educators that intend to teach a course on complexity theory. The book is also intended to promote interest in complexity theory and make it acccessible to general readers with adequate background (which is mainly being comfortable with abstract discussions, definitions and proofs). We expect most readers to have a basic knowledge of algorithms, or at least be fairly comfortable with the notion of an algorithm.The book focuses on several sub-areas of complexity theory (including, e.g., pseudorandomness and probabilistic proof systems). In each case, the exposition starts from the intuitive questions addresses by the sub-area, as embodied in the concepts that it studies. The exposition discusses the fundamental importance of these questions, the choices made in the actual formulation of these questions and notions, the approaches that underly the answers, and the ideas that are embedded in these answers. Our view is that these (
on-technical") aspects are the core of the field, and the presentation attempts to re ect this view.
552 citations
"The Complexity of Deciding Statisti..." refers background in this paper
...This could potentially make some problems easier, but there are complexity-theoretic results showing that several such problems are computationally hard, particularly when the input is a succinct description of a distribution....
[...]
TL;DR: In this paper, it was shown that the variation distance between the distribution of exchangeable random variables and the closest mixture of independent, identically distributed random variables is at most 2 ck/n, where c is the cardinality of the set.
Abstract: Let $X_1, X_2,\cdots, X_k, X_{k+1},\cdots, X_n$ be exchangeable random variables taking values in the set $S$. The variation distance between the distribution of $X_1, X_2,\cdots, X_k$ and the closest mixture of independent, identically distributed random variables is shown to be at most $2 ck/n$, where $c$ is the cardinality of $S$. If $c$ is infinite, the bound $k(k - 1)/n$ is obtained. These results imply the most general known forms of de Finetti's theorem. Examples are given to show that the rates $k/n$ and $k(k - 1)/n$ cannot be improved. The main tool is a bound on the variation distance between sampling with and without replacement. For instance, suppose an urn contains $n$ balls, each marked with some element of the set $S$, whose cardinality $c$ is finite. Now $k$ draws are made at random from this urn, either with or without replacement. This generates two probability distributions on the set of $k$-tuples, and the variation distance between them is at most $2 ck/n$.
461 citations
"The Complexity of Deciding Statisti..." refers background in this paper
...© 2015 Thomas Watson cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/toc.2015.v011a001...
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TL;DR: It turns out that some of the languages investigated for the succinct representation of the instances of combinatorial problems are not comparable, unless P=NP Some problems left open in [2].
Abstract: Several languages for the succinct representation of the instances of combinatorial problems are investigated. These languages have been introduced in [20, 2] and [5] where it has been shown that describing the instances by these languages causes a blow-up of the complexities of some problems. In the present paper the descriptional power of these languages is compared by estimating the complexities of some combinatorial problems in terms of completeness in suitable classes of the “counting polynomial-time hierarchy” which is introduced here. It turns out that some of the languages are not comparable, unless P=NP Some problems left open in [2] are solved.
321 citations
"The Complexity of Deciding Statisti..." refers background in this paper
...© 2015 Thomas Watson cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/toc.2015.v011a001...
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