56 research outputs found
Message passing and Monte Carlo algorithms: connecting fixed points with metastable states
Mean field-like approximations (including naive mean field, Bethe and Kikuchi
and more general Cluster Variational Methods) are known to stabilize ordered
phases at temperatures higher than the thermodynamical transition. For example,
in the Edwards-Anderson model in 2-dimensions these approximations predict a
spin glass transition at finite . Here we show that the spin glass solutions
of the Cluster Variational Method (CVM) at plaquette level do describe well
actual metastable states of the system. Moreover, we prove that these states
can be used to predict non trivial statistical quantities, like the
distribution of the overlap between two replicas. Our results support the idea
that message passing algorithms can be helpful to accelerate Monte Carlo
simulations in finite dimensional systems.Comment: 6 pages, 6 figure
Contamination Source Inference in Water Distribution Networks
We study the inference of the origin and the pattern of contamination in water distribution networks after the observation of contaminants in few nodes of the network
Replica Cluster Variational Method
We present a general formalism to make the Replica-Symmetric and
Replica-Symmetry-Breaking ansatz in the context of Kikuchi's Cluster
Variational Method (CVM). Using replicas and the message-passing formulation of
CVM we obtain a variational expression of the replicated free energy of a
system with quenched disorder, both averaged and on a single sample, and make
the hierarchical ansatz using functionals of functions of fields to represent
the messages. We begin to study the method considering the plaquette
approximation to the averaged free energy of the Edwards-Anderson model in the
paramagnetic Replica-Symmetric phase. In two dimensions we find that the
spurious spin-glass phase transition of the Bethe approximation disappears and
the paramagnetic phase is stable down to zero temperature in all the three
regular 2D lattices. The quantitative estimates of the free energy and of
various other quantities improve those of the Bethe approximation. We provide
the physical interpretation of the beliefs in the replica-symmetric phase as
disorder distributions of the local Hamiltonians. The messages instead do not
admit such an interpretation and indeed they cannot be represented as
populations in the spin-glass phase at variance with the Bethe approximation.Comment: 32 pages, 14 figures. In the revised version the content has been
reorganized in order to imporve readabilit
Characterizing and Improving Generalized Belief Propagation Algorithms on the 2D Edwards-Anderson Model
We study the performance of different message passing algorithms in the two
dimensional Edwards Anderson model. We show that the standard Belief
Propagation (BP) algorithm converges only at high temperature to a paramagnetic
solution. Then, we test a Generalized Belief Propagation (GBP) algorithm,
derived from a Cluster Variational Method (CVM) at the plaquette level. We
compare its performance with BP and with other algorithms derived under the
same approximation: Double Loop (DL) and a two-ways message passing algorithm
(HAK). The plaquette-CVM approximation improves BP in at least three ways: the
quality of the paramagnetic solution at high temperatures, a better estimate
(lower) for the critical temperature, and the fact that the GBP message passing
algorithm converges also to non paramagnetic solutions. The lack of convergence
of the standard GBP message passing algorithm at low temperatures seems to be
related to the implementation details and not to the appearance of long range
order. In fact, we prove that a gauge invariance of the constrained CVM free
energy can be exploited to derive a new message passing algorithm which
converges at even lower temperatures. In all its region of convergence this new
algorithm is faster than HAK and DL by some orders of magnitude.Comment: 19 pages, 13 figure
Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
We present and solve the Replica Symmetric equations in the context of the
Replica Cluster Variational Method for the 2D random bond Ising model
(including the 2D Edwards-Anderson spin glass model). First we solve a
linearized version of these equations to obtain the phase diagrams of the model
on the square and triangular lattices. In both cases the spin-glass transition
temperatures and the tricritical point estimations improve largely over the
Bethe predictions. Moreover, we show that this phase diagram is consistent with
the behavior of inference algorithms on single instances of the problem.
Finally, we present a method to consistently find approximate solutions to the
equations in the glassy phase. The method is applied to the triangular lattice
down to T=0, also in the presence of an external field.Comment: 22 pages, 11 figure
Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices
We solve the Edwards-Anderson model (EA) in different Husimi lattices. We
show that, at T=0, the structure of the solution space depends on the parity of
the loop sizes. Husimi lattices with odd loop sizes have always a trivial
paramagnetic solution stable under 1RSB perturbations while, in Husimi lattices
with even loop sizes, this solution is absent. The range of stability under
1RSB perturbations of this and other RS solutions is computed analytically
(when possible) or numerically. We compute the free-energy, the complexity and
the ground state energy of different Husimi lattices at the level of the 1RSB
approximation. We also show, when the fraction of ferromagnetic couplings
increases, the existence, first, of a discontinuous transition from a
paramagnetic to a spin glass phase and latter of a continuous transition from a
spin glass to a ferromagnetic phase.Comment: 20 pages, 10 figures (v3: Corrected analysis of transitions. Appendix
proof fixed
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
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