Sequential Cavity Method for Computing Free Energy and Surface Pressure

Gamarnik, David; Katz, Dmitriy

doi:10.1007/s10955-009-9849-3

Cited by 53 publications

(61 citation statements)

References 31 publications

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“…Then we adapt the "strip method" used for lattice shifts [11,13] to study the entropy h (k) of the golden mean subshift on the regular k-tree. Generalizing and improving the result in [14], we show in Theorem 3.7 that h (k) is strictly increasing in k. This contrasts with the apparent decrease of the entropy for the golden mean SFT's on Z k for k = 1, 2, 3, 4 [7]. In Theorems 3.1 and 4.1 we show that for each fixed k = 2, .…”

Section: Introductionsupporting

confidence: 60%

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Entropy on regular trees

Petersen¹,

Salama²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is Z. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on Z 2 , we show that the entropy of the hard square tree shift on the regular k-tree increases with k, in contrast to the case of Z k . We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for k = 2, . . . , 8 and propose that this holds for all k ≥ 2. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the k-tree for all k. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

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Section: Introductionsupporting

confidence: 60%

“…The entropy of the hard square tree shift on the k-tree can increase with the dimension k of the tree (in contrast with what is known about the hard square SFT on the lattice Z k[7]). In particular, we have…”

mentioning

confidence: 84%

Entropy on regular trees

Petersen¹,

Salama²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…A possible extension is mentioned below in Question 8. Note for comparison that numerical evidence [17] indicates that for the golden mean SFT's on the integer lattices Z d (also called the hard square or hard core models) the topological entropy seems to decrease with d = 1, 2, 3, 4. Proof.…”

Section: Thusmentioning

confidence: 99%

Tree shift topological entropy

Petersen

Salama

2018

Theoretical Computer Science

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“…The approach was also tested numerically on a grid graphs (finite subgraphs of a lattice Z d ) in Gamarnik and Katz [12], which resulted in significantly improved bounds on the underlying partition function, compared to some earlier numerical estimates based on other techniques. It would be of interest to develop a similar tight bound for the case of general λ i and µ i , but we are not aware of such results in the literature.…”

Section: Tutorials In Operations Research C 2013 Informsmentioning

confidence: 99%

Correlation Decay Method for Decision, Optimization, and Inference in Large-Scale Networks

Gamarnik¹

2013

Theory Driven by Influential Applications

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Please scroll down for article-it is on subsequent pagesINFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org Correlation Decay Method for Decision, Optimization, and Inference in Large-Scale Networks David GamarnikMassachusetts Institute of Technology, Cambridge, Massachusetts 02140, gamarnik@mit.edu Abstract Many inference and decision problems on networks involve solving hard algorithmic problems on graphs. Even when the underlying problem admits a polynomial time algorithm, such an algorithm can be impractical simply because of the sheer size of the underlying network, as many technological, communication, and natural networks may easily be of the order of millions or even billions of nodes. Thus one often has to resort to faster distributed (local) algorithms, which can be run in parallel on the underlying networks, providing the required time and computation effort scales. The correlation decay property, which can be established for a broad class of networks exhibiting random topology, random costs, or randomized decisions, is a tool that provides a rigorous justification for the validity of such local algorithms. In this tutorial we illustrate algorithmic methods based on the correlation decay property on several examples. Starting with the PageRank algorithm for search and navigation on the Internet, we will proceed to applications of the correlation decay method to problems in wireless communication, combinatorial optimization problems on graphs, and, finally, to the problem of network learning-a problem in the machine learning field.

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Sequential Cavity Method for Computing Free Energy and Surface Pressure

Cited by 53 publications

References 31 publications

Entropy on regular trees

Entropy on regular trees

Tree shift topological entropy

Correlation Decay Method for Decision, Optimization, and Inference in Large-Scale Networks

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