198 research outputs found
Survey propagation for the cascading Sourlas code
We investigate how insights from statistical physics, namely survey
propagation, can improve decoding of a particular class of sparse error
correcting codes. We show that a recently proposed algorithm, time averaged
belief propagation, is in fact intimately linked to a specific survey
propagation for which Parisi's replica symmetry breaking parameter is set to
zero, and that the latter is always superior to belief propagation in the high
connectivity limit. We briefly look at further improvements available by going
to the second level of replica symmetry breaking.Comment: 14 pages, 5 figure
Analysis of common attacks in LDPCC-based public-key cryptosystems
We analyze the security and reliability of a recently proposed class of
public-key cryptosystems against attacks by unauthorized parties who have
acquired partial knowledge of one or more of the private key components and/or
of the plaintext. Phase diagrams are presented, showing critical partial
knowledge levels required for unauthorized decryptionComment: 14 pages, 6 figure
Perceptron capacity revisited: classification ability for correlated patterns
In this paper, we address the problem of how many randomly labeled patterns
can be correctly classified by a single-layer perceptron when the patterns are
correlated with each other. In order to solve this problem, two analytical
schemes are developed based on the replica method and Thouless-Anderson-Palmer
(TAP) approach by utilizing an integral formula concerning random rectangular
matrices. The validity and relevance of the developed methodologies are shown
for one known result and two example problems. A message-passing algorithm to
perform the TAP scheme is also presented
Dynamical replica analysis of disordered Ising spin systems on finitely connected random graphs
We study the dynamics of macroscopic observables such as the magnetization
and the energy per degree of freedom in Ising spin models on random graphs of
finite connectivity, with random bonds and/or heterogeneous degree
distributions. To do so we generalize existing implementations of dynamical
replica theory and cavity field techniques to systems with strongly disordered
and locally tree-like interactions. We illustrate our results via application
to the dynamics of e.g. spin-glasses on random graphs and of the
overlap in finite connectivity Sourlas codes. All results are tested against
Monte Carlo simulations.Comment: 4 pages, 14 .eps file
Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
Compressed sensing is a signal processing method that acquires data directly
in a compressed form. This allows one to make less measurements than what was
considered necessary to record a signal, enabling faster or more precise
measurement protocols in a wide range of applications. Using an
interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a
strategy that allows compressed sensing to be performed at acquisition rates
approaching to the theoretical optimal limits. In this paper, we give a more
thorough presentation of our approach, and introduce many new results. We
present the probabilistic approach to reconstruction and discuss its optimality
and robustness. We detail the derivation of the message passing algorithm for
reconstruction and expectation max- imization learning of signal-model
parameters. We further develop the asymptotic analysis of the corresponding
phase diagrams with and without measurement noise, for different distribution
of signals, and discuss the best possible reconstruction performances
regardless of the algorithm. We also present new efficient seeding matrices,
test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe
Optimal Resource Allocation in Random Networks with Transportation Bandwidths
We apply statistical physics to study the task of resource allocation in
random sparse networks with limited bandwidths for the transportation of
resources along the links. Useful algorithms are obtained from recursive
relations. Bottlenecks emerge when the bandwidths are small, causing an
increase in the fraction of idle links. For a given total bandwidth per node,
the efficiency of allocation increases with the network connectivity. In the
high connectivity limit, we find a phase transition at a critical bandwidth,
above which clusters of balanced nodes appear, characterised by a profile of
homogenized resource allocation similar to the Maxwell's construction.Comment: 28 pages, 11 figure
Inference algorithms for gene networks: a statistical mechanics analysis
The inference of gene regulatory networks from high throughput gene
expression data is one of the major challenges in systems biology. This paper
aims at analysing and comparing two different algorithmic approaches. The first
approach uses pairwise correlations between regulated and regulating genes; the
second one uses message-passing techniques for inferring activating and
inhibiting regulatory interactions. The performance of these two algorithms can
be analysed theoretically on well-defined test sets, using tools from the
statistical physics of disordered systems like the replica method. We find that
the second algorithm outperforms the first one since it takes into account
collective effects of multiple regulators
Exact solutions for diluted spin glasses and optimization problems
We study the low temperature properties of p-spin glass models with finite
connectivity and of some optimization problems. Using a one-step functional
replica symmetry breaking Ansatz we can solve exactly the saddle-point
equations for graphs with uniform connectivity. The resulting ground state
energy is in perfect agreement with numerical simulations. For fluctuating
connectivity graphs, the same Ansatz can be used in a variational way: For
p-spin models (known as p-XOR-SAT in computer science) it provides the exact
configurational entropy together with the dynamical and static critical
connectivities (for p=3, \gamma_d=0.818 and \gamma_s=0.918 resp.), whereas for
hard optimization problems like 3-SAT or Bicoloring it provides new upper
bounds for their critical thresholds (\gamma_c^{var}=4.396 and
\gamma_c^{var}=2.149 resp.).Comment: 4 pages, 1 figure, accepted for publication in PR
On analyticity with respect to the replica number in random energy models I: an exact expression of the moment of the partition function
We provide an exact expression of the moment of the partition function for
random energy models of finite system size, generalizing an earlier expression
for a grand canonical version of the discrete random energy model presented by
the authors in Prog. Theor. Phys. 111, 661 (2004). The expression can be
handled both analytically and numerically, which is useful for examining how
the analyticity of the moment with respect to the replica numbers, which play
the role of powers of the moment, can be broken in the thermodynamic limit. A
comparison with a replica method analysis indicates that the analyticity
breaking can be regarded as the origin of the one-step replica symmetry
breaking. The validity of the expression is also confirmed by numerical methods
for finite systems.Comment: 16 pages, 4 figure
Typical performance of low-density parity-check codes over general symmetric channels
Typical performance of low-density parity-check (LDPC) codes over a general
binary-input output-symmetric memoryless channel is investigated using methods
of statistical mechanics. Theoretical framework for dealing with general
symmetric channels is provided, based on which Gallager and MacKay-Neal codes
are studied as examples of LDPC codes. It has been shown that the basic
properties of these codes known for particular channels, including the property
to potentially saturate Shannon's limit, hold for general symmetric channels.
The binary-input additive-white-Gaussian-noise channel and the binary-input
Laplace channel are considered as specific channel noise models.Comment: 10 pages, 4 figures, RevTeX4; an error in reference correcte
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