10,466 research outputs found
The condensation phase transition in the regular -SAT model
Much of the recent work on random constraint satisfaction problems has been
inspired by ingenious but non-rigorous approaches from physics. The physics
predictions typically come in the form of distributional fixed point problems
that are intended to mimic Belief Propagation, a message passing algorithm,
applied to the random CSP. In this paper we propose a novel method for
harnessing Belief Propagation directly to obtain a rigorous proof of such a
prediction, namely the existence and location of a condensation phase
transition in the random regular -SAT model.Comment: Revised version based on arXiv:1504.03975, version
One-step replica symmetry breaking of random regular NAE-SAT I
In a broad class of sparse random constraint satisfaction problems(CSP), deep
heuristics from statistical physics predict that there is a condensation phase
transition before the satisfiability threshold, governed by one-step replica
symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of
such random CSPs, it was verified \cite{ssz16} that its free energy is
well-defined and the explicit value follows the 1RSB prediction. However, for
any model of sparse random CSP, it has been unknown whether the solution space
indeed condensates on O(1) clusters according to the 1RSB prediction. In this
paper, we give an affirmative answer to this question for the random regular
k-NAE-SAT model. Namely, we prove that with probability bounded away from zero,
most of the solutions lie inside a bounded number of solution clusters whose
sizes are comparable to the scale of the free energy. Furthermore, we establish
that the overlap between two independently drawn solutions concentrates
precisely at two values. Our proof is based on a detailed moment analysis of a
spin system, which has an infinite spin space that encodes the structure of
solution clusters. We believe that our method is applicable to a broad range of
random CSPs in the 1RSB universality class.Comment: The previous version is divided into two parts and this submission is
Part I of a two-paper serie
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
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
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method
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