188 research outputs found
Approximating the Permanent with Fractional Belief Propagation
We discuss schemes for exact and approximate computations of permanents, and
compare them with each other. Specifically, we analyze the Belief Propagation
(BP) approach and its Fractional Belief Propagation (FBP) generalization for
computing the permanent of a non-negative matrix. Known bounds and conjectures
are verified in experiments, and some new theoretical relations, bounds and
conjectures are proposed. The Fractional Free Energy (FFE) functional is
parameterized by a scalar parameter , where
corresponds to the BP limit and corresponds to the exclusion
principle (but ignoring perfect matching constraints) Mean-Field (MF) limit.
FFE shows monotonicity and continuity with respect to . For every
non-negative matrix, we define its special value to be the
for which the minimum of the -parameterized FFE functional is
equal to the permanent of the matrix, where the lower and upper bounds of the
-interval corresponds to respective bounds for the permanent. Our
experimental analysis suggests that the distribution of varies for
different ensembles but always lies within the interval.
Moreover, for all ensembles considered the behavior of is highly
distinctive, offering an emprirical practical guidance for estimating
permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure
Palette-colouring: a belief-propagation approach
We consider a variation of the prototype combinatorial-optimisation problem
known as graph-colouring. Our optimisation goal is to colour the vertices of a
graph with a fixed number of colours, in a way to maximise the number of
different colours present in the set of nearest neighbours of each given
vertex. This problem, which we pictorially call "palette-colouring", has been
recently addressed as a basic example of problem arising in the context of
distributed data storage. Even though it has not been proved to be NP complete,
random search algorithms find the problem hard to solve. Heuristics based on a
naive belief propagation algorithm are observed to work quite well in certain
conditions. In this paper, we build upon the mentioned result, working out the
correct belief propagation algorithm, which needs to take into account the
many-body nature of the constraints present in this problem. This method
improves the naive belief propagation approach, at the cost of increased
computational effort. We also investigate the emergence of a satisfiable to
unsatisfiable "phase transition" as a function of the vertex mean degree, for
different ensembles of sparse random graphs in the large size ("thermodynamic")
limit.Comment: 22 pages, 7 figure
Markov entropy decomposition: a variational dual for quantum belief propagation
We present a lower bound for the free energy of a quantum many-body system at
finite temperature. This lower bound is expressed as a convex optimization
problem with linear constraints, and is derived using strong subadditivity of
von Neumann entropy and a relaxation of the consistency condition of local
density operators. The dual to this minimization problem leads to a set of
quantum belief propagation equations, thus providing a firm theoretical
foundation to that approach. The minimization problem is numerically tractable,
and we find good agreement with quantum Monte Carlo for the spin-half
Heisenberg anti-ferromagnet in two dimensions. This lower bound complements
other variational upper bounds. We discuss applications to Hamiltonian
complexity theory and give a generalization of the structure theorem of Hayden,
Jozsa, Petz and Winter to trees in an appendix
The Fully Frustrated Hypercubic Model is Glassy and Aging at Large
We discuss the behavior of the fully frustrated hypercubic cell in the
infinite dimensional mean-field limit. In the Ising case the system undergoes a
glass transition, well described by the random orthogonal model. Under the
glass temperature aging effects show clearly. In the case there is no sign
of a phase transition, and the system is always a paramagnet.Comment: Figures added in uufiles format, and epsf include
Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
We generalize the belief-propagation algorithm to sparse random networks with
arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in
these networks belongs to a given set of motifs (generalization of the
configuration model). These networks can be treated as sparse uncorrelated
hypergraphs in which hyperedges represent motifs. Here a hypergraph is a
generalization of a graph, where a hyperedge can connect any number of
vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which
crucially simplify the problem and allow us to apply the belief-propagation
algorithm to these loopy networks with arbitrary motifs. As natural examples,
we consider motifs in the form of finite loops and cliques. We apply the
belief-propagation algorithm to the ferromagnetic Ising model on the resulting
random networks. We obtain an exact solution of this model on networks with
finite loops or cliques as motifs. We find an exact critical temperature of the
ferromagnetic phase transition and demonstrate that with increasing the
clustering coefficient and the loop size, the critical temperature increases
compared to ordinary tree-like complex networks. Our solution also gives the
birth point of the giant connected component in these loopy networks.Comment: 9 pages, 4 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
Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices
We study the metastable states in Ising spin models with orthogonal
interaction matrices. We focus on three realizations of this model, the random
case and two non-random cases, i.e.\ the fully-frustrated model on an infinite
dimensional hypercube and the so-called sine-model. We use the mean-field (or
{\sc tap}) equations which we derive by resuming the high-temperature expansion
of the Gibbs free energy. In some special non-random cases, we can find the
absolute minimum of the free energy. For the random case we compute the average
number of solutions to the {\sc tap} equations. We find that the
configurational entropy (or complexity) is extensive in the range
T_{\mbox{\tiny RSB}}. Finally we present an apparently
unrelated replica calculation which reproduces the analytical expression for
the total number of {\sc tap} solutions.Comment: 22+3 pages, section 5 slightly modified, 1 Ref added, LaTeX and
uuencoded figures now independent of each other (easier to print). Postscript
available http://chimera.roma1.infn.it/index_papers_complex.htm
Locked constraint satisfaction problems
We introduce and study the random "locked" constraint satisfaction problems.
When increasing the density of constraints, they display a broad "clustered"
phase in which the space of solutions is divided into many isolated points.
While the phase diagram can be found easily, these problems, in their clustered
phase, are extremely hard from the algorithmic point of view: the best known
algorithms all fail to find solutions. We thus propose new benchmarks of really
hard optimization problems and provide insight into the origin of their typical
hardness.Comment: 4 pages, 2 figure
Computational core and fixed-point organisation in Boolean networks
In this paper, we analyse large random Boolean networks in terms of a
constraint satisfaction problem. We first develop an algorithmic scheme which
allows to prune simple logical cascades and under-determined variables,
returning thereby the computational core of the network. Second we apply the
cavity method to analyse number and organisation of fixed points. We find in
particular a phase transition between an easy and a complex regulatory phase,
the latter one being characterised by the existence of an exponential number of
macroscopically separated fixed-point clusters. The different techniques
developed are reinterpreted as algorithms for the analysis of single Boolean
networks, and they are applied to analysis and in silico experiments on the
gene-regulatory networks of baker's yeast (saccaromices cerevisiae) and the
segment-polarity genes of the fruit-fly drosophila melanogaster.Comment: 29 pages, 18 figures, version accepted for publication in JSTA
Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems
We review the understanding of the random constraint satisfaction problems,
focusing on the q-coloring of large random graphs, that has been achieved using
the cavity method of the physicists. We also discuss the properties of the
phase diagram in temperature, the connections with the glass transition
phenomenology in physics, and the related algorithmic issues.Comment: 10 pages, Proceedings of the International Workshop on
Statistical-Mechanical Informatics 2007, Kyoto (Japan) September 16-19, 200
- ā¦