15,341 research outputs found
Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models
The cluster variation method (CVM) is a hierarchy of approximate variational
techniques for discrete (Ising--like) models in equilibrium statistical
mechanics, improving on the mean--field approximation and the Bethe--Peierls
approximation, which can be regarded as the lowest level of the CVM. In recent
years it has been applied both in statistical physics and to inference and
optimization problems formulated in terms of probabilistic graphical models.
The foundations of the CVM are briefly reviewed, and the relations with
similar techniques are discussed. The main properties of the method are
considered, with emphasis on its exactness for particular models and on its
asymptotic properties.
The problem of the minimization of the variational free energy, which arises
in the CVM, is also addressed, and recent results about both provably
convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure
Loopy belief propagation and probabilistic image processing
Estimation of hyperparameters by maximization of the marginal likelihood in probabilistic image processing is investigated by using the cluster variation method. The algorithms are substantially equivalent to generalized loopy belief propagation
Cluster Expansion Method for Evolving Weighted Networks Having Vector-like Nodes
The Cluster Variation Method known in statistical mechanics and condensed
matter is revived for weighted bipartite networks. The decomposition of a
Hamiltonian through a finite number of components, whence serving to define
variable clusters, is recalled. As an illustration the network built from data
representing correlations between (4) macro-economic features, i.e. the so
called , of 15 EU countries, as (function) nodes, is
discussed. We show that statistical physics principles, like the maximum
entropy criterion points to clusters, here in a (4) variable phase space: Gross
Domestic Product (GDP), Final Consumption Expenditure (FCE), Gross Capital
Formation (GCF) and Net Exports (NEX). It is observed that the
entropy corresponds to a cluster which does explicitly include the GDP
but only the other (3) ''axes'', i.e. consumption, investment and trade
components. On the other hand, the entropy clustering scheme is
obtained from a coupling necessarily including GDP and FCE. The results confirm
intuitive economic theory and practice expectations at least as regards
geographical connexions. The technique can of course be applied to many other
cases in the physics of socio-economy networks.Comment: 7 pages, 2 figures, 20 references, 3 tables, submitted to FENS 07
Proceeding
Loop-corrected belief propagation for lattice spin models
Belief propagation (BP) is a message-passing method for solving probabilistic
graphical models. It is very successful in treating disordered models (such as
spin glasses) on random graphs. On the other hand, finite-dimensional lattice
models have an abundant number of short loops, and the BP method is still far
from being satisfactory in treating the complicated loop-induced correlations
in these systems. Here we propose a loop-corrected BP method to take into
account the effect of short loops in lattice spin models. We demonstrate,
through an application to the square-lattice Ising model, that loop-corrected
BP improves over the naive BP method significantly. We also implement
loop-corrected BP at the coarse-grained region graph level to further boost its
performance.Comment: 11 pages, minor changes with new references added. Final version as
published in EPJ
Deep learning systems as complex networks
Thanks to the availability of large scale digital datasets and massive
amounts of computational power, deep learning algorithms can learn
representations of data by exploiting multiple levels of abstraction. These
machine learning methods have greatly improved the state-of-the-art in many
challenging cognitive tasks, such as visual object recognition, speech
processing, natural language understanding and automatic translation. In
particular, one class of deep learning models, known as deep belief networks,
can discover intricate statistical structure in large data sets in a completely
unsupervised fashion, by learning a generative model of the data using
Hebbian-like learning mechanisms. Although these self-organizing systems can be
conveniently formalized within the framework of statistical mechanics, their
internal functioning remains opaque, because their emergent dynamics cannot be
solved analytically. In this article we propose to study deep belief networks
using techniques commonly employed in the study of complex networks, in order
to gain some insights into the structural and functional properties of the
computational graph resulting from the learning process.Comment: 20 pages, 9 figure
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