210 research outputs found
An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs
We first present an empirical study of the Belief Propagation (BP) algorithm,
when run on the random field Ising model defined on random regular graphs in
the zero temperature limit. We introduce the notion of maximal solutions for
the BP equations and we use them to fix a fraction of spins in their ground
state configuration. At the phase transition point the fraction of
unconstrained spins percolates and their number diverges with the system size.
This in turn makes the associated optimization problem highly non trivial in
the critical region. Using the bounds on the BP messages provided by the
maximal solutions we design a new and very easy to implement BP scheme which is
able to output a large number of stable fixed points. On one side this new
algorithm is able to provide the minimum energy configuration with high
probability in a competitive time. On the other side we found that the number
of fixed points of the BP algorithm grows with the system size in the critical
region. This unexpected feature poses new relevant questions on the physics of
this class of models.Comment: 20 pages, 8 figure
Survey propagation at finite temperature: application to a Sourlas code as a toy model
In this paper we investigate a finite temperature generalization of survey
propagation, by applying it to the problem of finite temperature decoding of a
biased finite connectivity Sourlas code for temperatures lower than the
Nishimori temperature. We observe that the result is a shift of the location of
the dynamical critical channel noise to larger values than the corresponding
dynamical transition for belief propagation, as suggested recently by
Migliorini and Saad for LDPC codes. We show how the finite temperature 1-RSB SP
gives accurate results in the regime where competing approaches fail to
converge or fail to recover the retrieval state
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
Statistical mechanics of optimization problems
Here I will present an introduction to the results that have been recently
obtained in constraint optimization of random problems using statistical
mechanics techniques. After presenting the general results, in order to
simplify the presentation I will describe in details the problems related to
the coloring of a random graph.Comment: proceedings of the conference SigmaPhi di Crete 2005, 10 pages, one
figur
Large deviations of cascade processes on graphs
Simple models of irreversible dynamical processes such as Bootstrap
Percolation have been successfully applied to describe cascade processes in a
large variety of different contexts. However, the problem of analyzing
non-typical trajectories, which can be crucial for the understanding of the
out-of-equilibrium phenomena, is still considered to be intractable in most
cases. Here we introduce an efficient method to find and analyze optimized
trajectories of cascade processes. We show that for a wide class of
irreversible dynamical rules, this problem can be solved efficiently on
large-scale systems
Local Message Passing on Frustrated Systems
Message passing on factor graphs is a powerful framework for probabilistic
inference, which finds important applications in various scientific domains.
The most wide-spread message passing scheme is the sum-product algorithm (SPA)
which gives exact results on trees but often fails on graphs with many small
cycles. We search for an alternative message passing algorithm that works
particularly well on such cyclic graphs. Therefore, we challenge the extrinsic
principle of the SPA, which loses its objective on graphs with cycles. We
further replace the local SPA message update rule at the factor nodes of the
underlying graph with a generic mapping, which is optimized in a data-driven
fashion. These modifications lead to a considerable improvement in performance
while preserving the simplicity of the SPA. We evaluate our method for two
classes of cyclic graphs: the 2x2 fully connected Ising grid and factor graphs
for symbol detection on linear communication channels with inter-symbol
interference. To enable the method for large graphs as they occur in practical
applications, we develop a novel loss function that is inspired by the Bethe
approximation from statistical physics and allows for training in an
unsupervised fashion.Comment: To appear at UAI 202
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