85,282 research outputs found
On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
We introduce a version of the cavity method for diluted mean-field spin
models that allows the computation of thermodynamic quantities similar to the
Franz-Parisi quenched potential in sparse random graph models. This method is
developed in the particular case of partially decimated random constraint
satisfaction problems. This allows to develop a theoretical understanding of a
class of algorithms for solving constraint satisfaction problems, in which
elementary degrees of freedom are sequentially assigned according to the
results of a message passing procedure (belief-propagation). We confront this
theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
The number of solutions for random regular NAE-SAT
Recent work has made substantial progress in understanding the transitions of
random constraint satisfaction problems. In particular, for several of these
models, the exact satisfiability threshold has been rigorously determined,
confirming predictions of statistical physics. Here we revisit one of these
models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is
natural to study, in the satisfiable regime, the number of solutions in a
typical instance. We prove here that these solutions have a well-defined free
energy (limiting exponential growth rate), with explicit value matching the
one-step replica symmetry breaking prediction. The proof develops new
techniques for analyzing a certain "survey propagation model" associated to
this problem. We believe that these methods may be applicable in a wide class
of related problems
Criticality and Heterogeneity in the Solution Space of Random Constraint Satisfaction Problems
Random constraint satisfaction problems are interesting model systems for
spin-glasses and glassy dynamics studies. As the constraint density of such a
system reaches certain threshold value, its solution space may split into
extremely many clusters. In this paper we argue that this ergodicity-breaking
transition is preceded by a homogeneity-breaking transition. For random K-SAT
and K-XORSAT, we show that many solution communities start to form in the
solution space as the constraint density reaches a critical value alpha_cm,
with each community containing a set of solutions that are more similar with
each other than with the outsider solutions. At alpha_cm the solution space is
in a critical state. The connection of these results to the onset of dynamical
heterogeneity in lattice glass models is discussed.Comment: 6 pages, 4 figures, final version as accepted by International
Journal of Modern Physics
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
Free Energy Subadditivity for Symmetric Random Hamiltonians
We consider a random Hamiltonian defined on a compact
space that admits a transitive action by a compact group .
When the law of is -invariant, we show its expected free energy
relative to the unique -invariant probability measure on
obeys a subadditivity property in the law of itself. The bound is often
tight for weak disorder and relates free energies at different temperatures
when is a Gaussian process. Many examples are discussed including branching
random walk, several spin glasses, random constraint satisfaction problems, and
the random field Ising model. We also provide a generalization to quantum
Hamiltonians with applications to the quantum SK and SYK models
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