Information-Theoretic Limits of Selecting Binary Graphical Models in High Dimensions

The problem of graphical model selection is to estimate the graph structure of a Markov random field given samples from it. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size and t...

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Vydáno v:	IEEE transactions on information theory Ročník 58; číslo 7; s. 4117 - 4134
Hlavní autoři:	Santhanam, Narayana P., Wainwright, Martin J.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.07.2012 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Analytical models Applied sciences Coding theory Coding, codes Decoding Exact sciences and technology Graphical models Graphs High dimensional inference Image edge detection Information theory Information, signal and communications theory KL divergence between Ising models Markov analysis Markov processes Markov random fields Probability Random variables sample complexity Sample size Signal and communications theory structure of Ising models Telecommunications and information theory Vectors Error probability Probabilistic approach Necessary and sufficient condition Model selection structure of Ising models Decoding Markov model Markov random fields sample complexity Random field High dimensional inference Pairwise Markov model KL divergence between Ising models Ising model Information theory
ISSN:	0018-9448, 1557-9654
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Popis
Shrnutí:	The problem of graphical model selection is to estimate the graph structure of a Markov random field given samples from it. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size and the number of edges k, and/or the maximal node degree d, are allowed to increase to infinity as a function of the sample size n. For pair-wise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class G p,k of graphs on vertices with at most k edges, and over the class G p,d of graphs on p vertices with maximum degree at most d. For the class G p,k , we establish the existence of constants c and c' such that if n <; ck log p, any method has error probability at least 1/2 uniformly over the family, and we demonstrate a graph decoder that succeeds with high probability uniformly over the family for sample sizes n >; c' k 2 log p. Similarly, for the class G p,d , we exhibit constants c and c' such that for n <; cd 2 log p, any method fails with probability at least 1/2, and we demonstrate a graph decoder that succeeds with high probability for n >; c' d 3 log p.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0018-9448 1557-9654
DOI:	10.1109/TIT.2012.2191659