10,790 research outputs found
Cycle-based Cluster Variational Method for Direct and Inverse Inference
We elaborate on the idea that loop corrections to belief propagation could be
dealt with in a systematic way on pairwise Markov random fields, by using the
elements of a cycle basis to define region in a generalized belief propagation
setting. The region graph is specified in such a way as to avoid dual loops as
much as possible, by discarding redundant Lagrange multipliers, in order to
facilitate the convergence, while avoiding instabilities associated to minimal
factor graph construction. We end up with a two-level algorithm, where a belief
propagation algorithm is run alternatively at the level of each cycle and at
the inter-region level. The inverse problem of finding the couplings of a
Markov random field from empirical covariances can be addressed region wise. It
turns out that this can be done efficiently in particular in the Ising context,
where fixed point equations can be derived along with a one-parameter log
likelihood function to minimize. Numerical experiments confirm the
effectiveness of these considerations both for the direct and inverse MRF
inference.Comment: 47 pages, 16 figure
The Grushko decomposition of a finite graph of finite rank free groups: an algorithm
A finitely generated group admits a decomposition, called its Grushko
decomposition, into a free product of freely indecomposable groups. There is an
algorithm to construct the Grushko decomposition of a finite graph of finite
rank free groups. In particular, it is possible to decide if such a group is
free.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper41.abs.htm
Symmetrically colored Gaussian graphical models with toric vanishing ideal
A colored Gaussian graphical model is a linear concentration model in which
equalities among the concentrations are specified by a coloring of an
underlying graph. The model is called RCOP if this coloring is given by the
edge and vertex orbits of a subgroup of the automorphism group of the graph. We
show that RCOP Gaussian graphical models on block graphs are toric in the space
of covariance matrices and we describe Markov bases for them. To this end, we
learn more about the combinatorial structure of these models and their
connection with Jordan algebras.Comment: Comments are very welcome
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