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Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels.

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Title: Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels.
Authors: Liu, Wenjian1 (AUTHOR) wjliu@qcc.cuny.edu, Ning, Ning2 (AUTHOR) ningnin@uw.edu
Source: Journal of Statistical Physics. Mar2019, Vol. 174 Issue 6, p1161-1188. 28p.
Subjects: IEEE Computer Society, Institute of Electrical & Electronics Engineers, Moments method (Statistics), Markov random fields, Information theory, Nonlinear dynamical systems, Statistical physics
Abstract: In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs et al. (in: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS, IEEE Computer Society, 2006), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs et al. (in: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS, IEEE Computer Society, 2006), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10955-019-02228-0
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      – SubjectFull: Moments method (Statistics)
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      – SubjectFull: Markov random fields
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              Text: Mar2019
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