120 research outputs found
Decimation flows in constraint satisfaction problems
We study hard constraint satisfaction problems with a decimation approach
based on message passing algorithms. Decimation induces a renormalization flow
in the space of problems, and we exploit the fact that this flow transforms
some of the constraints into linear constraints over GF(2). In particular, when
the flow hits the subspace of linear problems, one can stop decimation and use
Gaussian elimination. We introduce a new decimation algorithm which uses this
linear structure and shows a strongly improved performance with respect to the
usual decimation methods on some of the hardest locked occupation problems.Comment: 14 pages, 2 figure
Distribution of diameters for Erd\"os-R\'enyi random graphs
We study the distribution of diameters d of Erd\"os-R\'enyi random graphs
with average connectivity c. The diameter d is the maximum among all shortest
distances between pairs of nodes in a graph and an important quantity for all
dynamic processes taking place on graphs. Here we study the distribution P(d)
numerically for various values of c, in the non-percolating and the percolating
regime. Using large-deviations techniques, we are able to reach small
probabilities like 10^{-100} which allow us to obtain the distribution over
basically the full range of the support, for graphs up to N=1000 nodes. For
values c<1, our results are in good agreement with analytical results, proving
the reliability of our numerical approach. For c>1 the distribution is more
complex and no complete analytical results are available. For this parameter
range, P(d) exhibits an inflection point, which we found to be related to a
structural change of the graphs. For all values of c, we determined the
finite-size rate function Phi(d/N) and were able to extrapolate numerically to
N->infinity, indicating that the large deviation principle holds.Comment: 9 figure
Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning
We consider dictionary learning and blind calibration for signals and
matrices created from a random ensemble. We study the mean-squared error in the
limit of large signal dimension using the replica method and unveil the
appearance of phase transitions delimiting impossible, possible-but-hard and
possible inference regions. We also introduce an approximate message passing
algorithm that asymptotically matches the theoretical performance, and show
through numerical tests that it performs very well, for the calibration
problem, for tractable system sizes.Comment: 5 page
Reweighted belief propagation and quiet planting for random K-SAT
We study the random K-satisfiability problem using a partition function where
each solution is reweighted according to the number of variables that satisfy
every clause. We apply belief propagation and the related cavity method to the
reweighted partition function. This allows us to obtain several new results on
the properties of random K-satisfiability problem. In particular the
reweighting allows to introduce a planted ensemble that generates instances
that are, in some region of parameters, equivalent to random instances. We are
hence able to generate at the same time a typical random SAT instance and one
of its solutions. We study the relation between clustering and belief
propagation fixed points and we give a direct evidence for the existence of
purely entropic (rather than energetic) barriers between clusters in some
region of parameters in the random K-satisfiability problem. We exhibit, in
some large planted instances, solutions with a non-trivial whitening core; such
solutions were known to exist but were so far never found on very large
instances. Finally, we discuss algorithmic hardness of such planted instances
and we determine a region of parameters in which planting leads to satisfiable
benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression
correcte
Emergence of rigidity at the structural glass transition: a first principle computation
We compute the shear modulus of structural glasses from a first principle
approach based on the cloned liquid theory. We find that the intra-state
shear-modulus, which corresponds to the plateau modulus measured in linear
visco-elastic measurements, strongly depends on temperature and vanishes
continuously when the temperature is increased beyond the glass temperature.Comment: 5 pages, 2 figures. Revised version. An error in Fig 2 due to an
error in the plotting program is corrected. The revised figure is submitted
to PRL as an erratu
Spin glass theory and its new challenge: structured disorder
This paper first describes, from a high level viewpoint, the main challenges
that had to be solved in order to develop a theory of spin glasses in the last
fifty years. It then explains how important inference problems, notably those
occurring in machine learning, can be formulated as problems in statistical
physics of disordered systems. However, the main questions that we face in the
analysis of deep networks require to develop a new chapter of spin glass
theory, which will address the challenge of structured data.Comment: 17 pages, one figur
Mean-field message-passing equations in the Hopfield model and its generalizations
International audienceMotivated by recent progress in using restricted Boltzmann machines as preprocess-ing algorithms for deep neural network, we revisit the mean-field equations (belief-propagation and TAP equations) in the best understood such machine, namely the Hopfield model of neural networks, and we explicit how they can be used as iterative message-passing algorithms, providing a fast method to compute the local polariza-tions of neurons. In the "retrieval phase" where neurons polarize in the direction of one memorized pattern, we point out a major difference between the belief propagation and TAP equations : the set of belief propagation equations depends on the pattern which is retrieved, while one can use a unique set of TAP equations. This makes the latter method much better suited for applications in the learning process of restricted Boltzmann machines. In the case where the patterns memorized in the Hopfield model are not independent, but are correlated through a combinatorial structure, we show that the TAP equations have to be modified. This modification can be seen either as an alteration of the reaction term in TAP equations, or, more interestingly, as the consequence of message passing on a graphical model with several hidden layers, where the number of hidden layers depends on the depth of the correlations in the memorized patterns. This layered structure is actually necessary when one deals with more general restricted Boltzmann machines
How to compute the thermodynamics of a glass using a cloned liquid
The recently proposed strategy for studying the equilibrium thermodynamics of
the glass phase using a molecular liquid is reviewed and tested in details on
the solvable case of the -spin model. We derive the general phase diagram,
and confirm the validity of this procedure. We point out the efficacy of a
system of two weakly coupled copies in order to identify the glass transition,
and the necessity to study a system with copies ('clones') of the
original problem in order to derive the thermodynamic properties of the glass
phase.Comment: Latex, 17 pages, 6 figure
Dynamic message-passing equations for models with unidirectional dynamics
Understanding and quantifying the dynamics of disordered out-of-equilibrium
models is an important problem in many branches of science. Using the dynamic
cavity method on time trajectories, we construct a general procedure for
deriving the dynamic message-passing equations for a large class of models with
unidirectional dynamics, which includes the zero-temperature random field Ising
model, the susceptible-infected-recovered model, and rumor spreading models. We
show that unidirectionality of the dynamics is the key ingredient that makes
the problem solvable. These equations are applicable to single instances of the
corresponding problems with arbitrary initial conditions, and are
asymptotically exact for problems defined on locally tree-like graphs. When
applied to real-world networks, they generically provide a good analytic
approximation of the real dynamics.Comment: Final versio
Energy transport in strongly disordered superconductors and magnets
We develop an analytical theory for quantum phase transitions driven by
disorder in magnets and superconductors. We study these transitions with a
cavity approximation which becomes exact on a Bethe lattice with large
branching number. We find two different disordered phases, characterized by
very different relaxation rates, which both exhibit strong inhomogeneities
typical of glassy physics.Comment: 4 pages, 1 figur
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