853 research outputs found
Truncating the loop series expansion for Belief Propagation
Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the
partition sum (normalization constant) corresponding to a graphical model,
which is an expansion around the Belief Propagation solution. By adding
correction terms to the BP free energy, one for each "generalized loop" in the
factor graph, the exact partition sum is obtained. However, the usually
enormous number of generalized loops generally prohibits summation over all
correction terms. In this article we introduce Truncated Loop Series BP
(TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y.
Chernyak by considering generalized loops as compositions of simple loops. We
analyze the performance of TLSBP in different scenarios, including the Ising
model, regular random graphs and on Promedas, a large probabilistic medical
diagnostic system. We show that TLSBP often improves upon the accuracy of the
BP solution, at the expense of increased computation time. We also show that
the performance of TLSBP strongly depends on the degree of interaction between
the variables. For weak interactions, truncating the series leads to
significant improvements, whereas for strong interactions it can be
ineffective, even if a high number of terms is considered.Comment: 31 pages, 12 figures, submitted to Journal of Machine Learning
Researc
Belief Propagation and Loop Series on Planar Graphs
We discuss a generic model of Bayesian inference with binary variables
defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is
used to evaluate the resulting series expansion for the partition function. We
show that, for planar graphs, truncating the series at single-connected loops
reduces, via a map reminiscent of the Fisher transformation [3], to evaluating
the partition function of the dimer matching model on an auxiliary planar
graph. Thus, the truncated series can be easily re-summed, using the Pfaffian
formula of Kasteleyn [4]. This allows to identify a big class of
computationally tractable planar models reducible to a dimer model via the
Belief Propagation (gauge) transformation. The Pfaffian representation can also
be extended to the full Loop Series, in which case the expansion becomes a sum
of Pfaffian contributions, each associated with dimer matchings on an extension
to a subgraph of the original graph. Algorithmic consequences of the Pfaffian
representation, as well as relations to quantum and non-planar models, are
discussed.Comment: Accepted for publication in Journal of Statistical Mechanics: theory
and experimen
Loop Calculus for Non-Binary Alphabets using Concepts from Information Geometry
The Bethe approximation is a well-known approximation of the partition
function used in statistical physics. Recently, an equality relating the
partition function and its Bethe approximation was obtained for graphical
models with binary variables by Chertkov and Chernyak. In this equality, the
multiplicative error in the Bethe approximation is represented as a weighted
sum over all generalized loops in the graphical model. In this paper, the
equality is generalized to graphical models with non-binary alphabet using
concepts from information geometry.Comment: 18 pages, 4 figures, submitted to IEEE Trans. Inf. Theor
Finite size corrections to disordered systems on Erd\"{o}s-R\'enyi random graphs
We study the finite size corrections to the free energy density in disorder
spin systems on sparse random graphs, using both replica theory and cavity
method. We derive an analytical expressions for the corrections in the
replica symmetric phase as a linear combination of the free energies of open
and closed chains. We perform a numerical check of the formulae on the Random
Field Ising Model at zero temperature, by computing finite size corrections to
the ground state energy density.Comment: Submitted to PR
Sums over geometries and improvements on the mean field approximation
The saddle points of a Lagrangian due to Efetov are analyzed. This Lagrangian
was originally proposed as a tool for calculating systematic corrections to the
Bethe approximation, a mean-field approximation which is important in
statistical mechanics, glasses, coding theory, and combinatorial optimization.
Detailed analysis shows that the trivial saddle point generates a sum over
geometries reminiscent of dynamically triangulated quantum gravity, which
suggests new possibilities to design sums over geometries for the specific
purpose of obtaining improved mean field approximations to -dimensional
theories. In the case of the Efetov theory, the dominant geometries are locally
tree-like, and the sum over geometries diverges in a way that is similar to
quantum gravity's divergence when all topologies are included. Expertise from
the field of dynamically triangulated quantum gravity about sums over
geometries may be able to remedy these defects and fulfill the Efetov theory's
original promise. The other saddle points of the Efetov Lagrangian are also
analyzed; the Hessian at these points is nonnormal and pseudo-Hermitian, which
is unusual for bosonic theories. The standard formula for Gaussian integrals is
generalized to nonnormal kernels.Comment: Accepted for publication in Physical Review D, probably in November
2007. At the reviewer's request, material was added which made the article
more assertive, confident, and clear. No changes in substanc
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