3,660 research outputs found
The Complexity of Approximating a Bethe Equilibrium
This paper resolves a common complexity issue in the Bethe approximation of
statistical physics and the Belief Propagation (BP) algorithm of artificial
intelligence. The Bethe approximation and the BP algorithm are heuristic
methods for estimating the partition function and marginal probabilities in
graphical models, respectively. The computational complexity of the Bethe
approximation is decided by the number of operations required to solve a set of
non-linear equations, the so-called Bethe equation. Although the BP algorithm
was inspired and developed independently, Yedidia, Freeman and Weiss (2004)
showed that the BP algorithm solves the Bethe equation if it converges
(however, it often does not). This naturally motivates the following question
to understand limitations and empirical successes of the Bethe and BP methods:
is the Bethe equation computationally easy to solve? We present a
message-passing algorithm solving the Bethe equation in a polynomial number of
operations for general binary graphical models of n variables where the maximum
degree in the underlying graph is O(log n). Our algorithm can be used as an
alternative to BP fixing its convergence issue and is the first fully
polynomial-time approximation scheme for the BP fixed-point computation in such
a large class of graphical models, while the approximate fixed-point
computation is known to be (PPAD-)hard in general. We believe that our
technique is of broader interest to understand the computational complexity of
the cavity method in statistical physics
Bethe free-energy approximations for disordered quantum systems
Given a locally consistent set of reduced density matrices, we construct
approximate density matrices which are globally consistent with the local
density matrices we started from when the trial density matrix has a tree
structure. We employ the cavity method of statistical physics to find the
optimal density matrix representation by slowly decreasing the temperature in
an annealing algorithm, or by minimizing an approximate Bethe free energy
depending on the reduced density matrices and some cavity messages originated
from the Bethe approximation of the entropy. We obtain the classical Bethe
expression for the entropy within a naive (mean-field) approximation of the
cavity messages, which is expected to work well at high temperatures. In the
next order of the approximation, we obtain another expression for the Bethe
entropy depending only on the diagonal elements of the reduced density
matrices. In principle, we can improve the entropy approximation by considering
more accurate cavity messages in the Bethe approximation of the entropy. We
compare the annealing algorithm and the naive approximation of the Bethe
entropy with exact and approximate numerical simulations for small and large
samples of the random transverse Ising model on random regular graphs.Comment: 23 pages, 4 figures, 4 appendice
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
Loop corrections in spin models through density consistency
Computing marginal distributions of discrete or semidiscrete Markov random
fields (MRFs) is a fundamental, generally intractable problem with a vast
number of applications in virtually all fields of science. We present a new
family of computational schemes to approximately calculate the marginals of
discrete MRFs. This method shares some desirable properties with belief
propagation, in particular, providing exact marginals on acyclic graphs, but it
differs with the latter in that it includes some loop corrections; i.e., it
takes into account correlations coming from all cycles in the factor graph. It
is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs
with the latter in that the consistency is not on the first two moments of the
distribution but rather on the value of its density on a subset of values. The
results on finite-dimensional Isinglike models show a significant improvement
with respect to the Bethe-Peierls (tree) approximation in all cases and with
respect to the plaquette cluster variational method approximation in many
cases. In particular, for the critical inverse temperature of the
homogeneous hypercubic lattice, the expansion of
around of the proposed scheme is exact up to the order,
whereas the two latter are exact only up to the order.Comment: 12 pages, 3 figures, 1 tabl
Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
Transfer matrices and matrix product operators play an ubiquitous role in the
field of many body physics. This paper gives an ideosyncratic overview of
applications, exact results and computational aspects of diagonalizing transfer
matrices and matrix product operators. The results in this paper are a mixture
of classic results, presented from the point of view of tensor networks, and of
new results. Topics discussed are exact solutions of transfer matrices in
equilibrium and non-equilibrium statistical physics, tensor network states,
matrix product operator algebras, and numerical matrix product state methods
for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit
Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling
We present a general framework how to investigate stability of solutions
within a single self-consistent renormalization scheme being a parquet-type
extension of the Baym-Kadanoff construction of conserving approximations. To
obtain a consistent description of one- and two-particle quantities, needed for
the stability analysis, we impose equations of motion on the one- as well on
the two-particle Green functions simultaneously and introduce approximations in
their input, the completely irreducible two-particle vertex. Thereby we do not
loose singularities caused by multiple two-particle scatterings. We find a
complete set of stability criteria and show that each instability, singularity
in a two-particle function, is connected with a symmetry-breaking order
parameter, either of density type or anomalous. We explicitly study the Hubbard
model at intermediate coupling and demonstrate that approximations with static
vertices get unstable before a long-range order or a metal-insulator transition
can be reached. We use the parquet approximation and turn it to a workable
scheme with dynamical vertex corrections. We derive a qualitatively new theory
with two-particle self-consistence, the complexity of which is comparable with
FLEX-type approximations. We show that it is the simplest consistent and stable
theory being able to describe qualitatively correctly quantum critical points
and the transition from weak to strong coupling in correlated electron systems.Comment: REVTeX, 26 pages, 12 PS figure
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