arXiv subjects

Michael Chertkov

Publications and source records attributed to Michael Chertkov.

178 records · Page 10Linked to original sources

arXiv2007

An Efficient Pseudo-Codeword Search Algorithm for Linear Programming Decoding of LDPC Codes

In Linear Programming (LP) decoding of a Low-Density-Parity-Check (LDPC) code one minimizes a linear functional, with coefficients related to log-likelihood ratios, over a relaxation of the polytope spanned by the codewords \cite{03FWK}. In order to quantify LP decoding, and thus to describe performance of the error-correction scheme at moderate and large Signal-to-Noise-Ratios (SNR), it is important to study the relaxed polytope to understand better its vertexes, so-called pseudo-codewords, especially those which are neighbors of the zero codeword. In this manuscript we propose a technique to heuristically create a list of these neighbors and their distances. Our pseudo-codeword-search algorithm starts by randomly choosing the initial configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two sub-steps. Firstly, one applies an LP decoder to the noise-configuration deriving a pseudo-codeword. Secondly, one finds configuration of the noise equidistant from the pseudo codeword and the zero codeword. The resulting noise configuration is used as an entry for the next step. The iterations converge rapidly to a pseudo-codeword neighboring the zero codeword. Repeated many times, this procedure is characterized by the distribution function (frequency spectrum) of the pseudo-codeword effective distance. The effective distance of the coding scheme is approximated by the shortest distance pseudo-codeword in the spectrum. The efficiency of the procedure is demonstrated on examples of the Tanner $[155,64,20]$ code and Margulis $p=7$ and $p=11$ codes (672 and 2640 bits long respectively) operating over an Additive-White-Gaussian-Noise (AWGN) channel.

cs.IT↗

arXiv2007

Loop series for discrete statistical models on graphs

In this paper we present derivation details, logic, and motivation for the loop calculus introduced in \cite{06CCa}. Generating functions for three inter-related discrete statistical models are each expressed in terms of a finite series. The first term in the series corresponds to the Bethe-Peierls (Belief Propagation)-BP contribution, the other terms are labeled by loops on the factor graph. All loop contributions are simple rational functions of spin correlation functions calculated within the BP approach. We discuss two alternative derivations of the loop series. One approach implements a set of local auxiliary integrations over continuous fields with the BP contribution corresponding to an integrand saddle-point value. The integrals are replaced by sums in the complimentary approach, briefly explained in \cite{06CCa}. A local gauge symmetry transformation that clarifies an important invariant feature of the BP solution, is revealed in both approaches. The partition function remains invariant while individual terms change under the gauge transformation. The requirement for all individual terms to be non-zero only for closed loops in the factor graph (as opposed to paths with loose ends) is equivalent to fixing the first term in the series to be exactly equal to the BP contribution. Further applications of the loop calculus to problems in statistical physics, computer and information sciences are discussed.

cond-mat.stat-mech↗

arXiv2007

Reducing the Error Floor

We discuss how the loop calculus approach of [Chertkov, Chernyak '06], enhanced by the pseudo-codeword search algorithm of [Chertkov, Stepanov '06] and the facet-guessing idea from [Dimakis, Wainwright '06], improves decoding of graph based codes in the error-floor domain. The utility of the new, Linear Programming based, decoding is demonstrated via analysis and simulations of the model $[155,64,20]$ code.

cs.IT↗

arXiv2007

Pseudo-codeword Landscape

We discuss the performance of Low-Density-Parity-Check (LDPC) codes decoded by means of Linear Programming (LP) at moderate and large Signal-to-Noise-Ratios (SNR). Utilizing a combination of the previously introduced pseudo-codeword-search method and a new "dendro" trick, which allows us to reduce the complexity of the LP decoding, we analyze the dependence of the Frame-Error-Rate (FER) on the SNR. Under Maximum-A-Posteriori (MAP) decoding the dendro-code, having only checks with connectivity degree three, performs identically to its original code with high-connectivity checks. For a number of popular LDPC codes performing over the Additive-White-Gaussian-Noise (AWGN) channel we found that either an error-floor sets at a relatively low SNR, or otherwise a transient asymptote, characterized by a faster decay of FER with the SNR increase, precedes the error-floor asymptote. We explain these regimes in terms of the pseudo-codeword spectra of the codes.

cs.IT↗

arXiv2006

Loop Calculus Helps to Improve Belief Propagation and Linear Programming Decodings of Low-Density-Parity-Check Codes

We illustrate the utility of the recently developed loop calculus for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code. The log-likelihood specific critical loop is found by means of the loop calculus. The general method is tested using an example of the Linear Programming (LP) decoding, that can be viewed as a special limit of the BP decoding. Considering the (155,64,20) code that performs over Additive-White-Gaussian-Noise channel we show that the loop calculus improves the LP decoding and corrects all previously found dangerous configurations of log-likelihoods related to pseudo-codewords with low effective distance, thus reducing the code's error-floor.

cs.IT↗

arXiv2006

Path-integral analysis of fluctuation theorems for general Langevin processes

We examine classical, transient fluctuation theorems within the unifying framework of Langevin dynamics. We explicitly distinguish between the effects of non-conservative forces that violate detailed balance, and non-autonomous dynamics arising from the variation of an external parameter. When both these sources of nonequilibrium behavior are present, there naturally arise two distinct fluctuation theorems.

cond-mat.stat-mech↗

arXiv2006

Loop Calculus in Statistical Physics and Information Science

Considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series. The leading term in the series is the Bethe-Peierls (Belief Propagation)-BP contribution, the rest are expressed as loop-contributions on the factor graph and calculated directly using the BP solution. The series unveils a small parameter that often makes the BP approximation so successful. Applications of the loop calculus in statistical physics and information science are discussed.

cond-mat.stat-mech↗

arXiv2003

Periodic and Quasi-Periodic Compensation Strategies of Extreme Outages caused by Polarization Mode Dispersion and Amplifier Noise

Effect of birefringent disorder on the Bit Error Rate (BER) in an optical fiber telecommunication system subject to amplifier noise may lead to extreme outages, related to anomalously large values of BER. We analyze the Probability Distribution Function (PDF) of BER for various strategies of Polarization Mode Dispersion (PMD) compensation. A compensation method is proposed that is capable of more efficient extreme outages suppression, which leads to substantial improvement of the fiber system performance.

physics.optics↗

arXiv2003

Compensation for Extreme Outages caused by Polarization Mode Dispersion and Amplifier noise

Joint effect of weak birefringent disorder and amplifier noise on transmission in optical fiber communication systems appears to be strong. The probability of an extreme outage that corresponds to anomalously large values of Bit Error Rate (BER) is perceptible. We analyze the dependence of the Probability Distribution Function (PDF) of BER on the first-order and also higher-order PMD compensation schemes.

physics.optics↗

arXiv2003

Extreme Outages due to Polarization Mode Dispersion

We investigate the dependence of the bit-error-rate (BER) caused by amplifier noise in a linear optical fiber telecommunication system on the fiber birefringence. We show that the probability distribution function (PDF) of BER obtained by averaging over many realizations of birefringent disorder has an extended tail corresponding to anomalously large values of BER. We specifically discuss the dependence of the tail on such details of the pulse detection at the fiber output as "setting the clock" and filtering procedures.

physics.optics↗

arXiv2003

Probability of anomalously large Bit-Error-Rate in long haul optical transmission

We consider a linear model of optical pulse transmission through fiber with birefringent disorder in the presence of amplifier noise. Both disorder and noise are assumed to be weak, i.e. the average bit-error rate (BER) is small. The probability of rare violent events leading to the values of BER much larger than its typical value is estimated. We show that the probability distribution has a long algebraic tail.

cond-mat.stat-mech↗

arXiv2000

Polymer Stretching by Turbulence

The stretching of a polymer chain by a large scale chaotic flow is considered. The steady state which emerges as a balance of the turbulent stretching and anharmonic resistance of the chain is quantitatively described, i.e. the dependency on the flow parameters (Lyapunov exponent statistics) and the chain characteristics (the number of beads and the inter-bead elastic potential) is made explicit.

chao-dyn↗

arXiv1999

Lagrangian Tetrad Dynamics and the Phenomenology of Turbulence

A new phenomenological model of turbulent fluctuations is constructed by considering the Lagrangian dynamics of 4 points (the tetrad). The closure of the equations of motion is achieved by postulating an anisotropic, i.e. tetrad shape dependent, relation of the local pressure and the velocity gradient defined on the tetrad. The non-local contribution to the pressure and the incoherent small scale fluctuations are modeled as Gaussian white "noise". The resulting stochastic model for the coarse-grained velocity gradient is analyzed approximately, yielding predictions for the probability distribution functions of different 2nd and 3d order invariants. The results are compared with the direct numerical simulation of the Navier-Stokes. The model provides a reasonable representation of the non-linear dynamics involved in energy transfer and vortex stretching and allows to study interesting aspects of the statistical geometry of turbulence, e.g. vorticity/strain alignment. In a state with a constant energy flux (and K41 power spectrum) it exhibits the anomalous scaling of high moments associated with formation of high gradient sheets - events associated with large energy transfer. An approach to the more complete analysis of the stochastic model, properly including the effect of fluctuations, is outlined and will enable further quantitative juxtaposition of the model with the results of the DNS.

chao-dyn↗

arXiv1998

Passive advection in nonlinear medium

Forced advection of passive tracer, $θ$, in nonlinear relaxational medium by large scale (Batchelor problem) incompressible velocity field at scales less than the correlation length of the flow and larger than the diffusion scale is considered. Effective theory explaining small scale scalar fluctuations is proven to be linear, asymptotic free (downscales from the scale of the pumping) and universal. Only three parameters are required to decribe exhaustively the small scale statistics of scalar difference: two velocity-dependent ones, average and dispersion ($\barλ$ and $Δ$ respectively) of the exponential stretching rate of a trial line element, and $α$, standing for average rate of linear damping of small scale scalar fluctuations. $α$ is an explicit functional of potential chracterized medium nonlinearity and amplitude of $θ^{2}$ flux pumped into the system. Structure functions show an extremely anomalous, intermittent behavior: $<|δθ_{r}|^{q}> \sim r^{ξ_{q}}, ξ_{q} = \min {q,\sqrt{[ \frac{\barλ}Δ] ^{2} + \frac{2αq}Δ} - \frac{\barλ}Δ}$. No dissipative anomaly is found in the problem.

chao-dyn↗

arXiv1998

On how a joint interaction of two innocent partners (smooth advection & linear damping) produces a strong intermittency

Forced advection of passive scalar by a smooth $d$-dimensional incompressible velocity in the presence of a linear damping is studied. Acting separately advection and dumping do not lead to an essential intermittency of the steady scalar statistics, while being mixed together produce a very strong non-Gaussianity in the convective range: $q$-th (positive) moment of the absolute value of scalar difference, $<|θ(t;{\bf r})-θ(t;0)|^{q}> $ is proportional to $r^{ξ_{q}}$, $ξ_{q}=\sqrt{d^{2}/4+αdq/[ (d-1)D]}-d/2$, where $α/D$ measures the rate of the damping in the units of the stretching rate. Probability density function (PDF) of the scalar difference is also found.

chao-dyn↗

arXiv1994

Long-time dynamics of the infinite-temperature Heisenberg magnet

Infinite-temperature long-time dynamics of Heisenberg model ${\bf\hat{H}}=-\frac{1}{2}\sum_{i,j}J_{ij} \hat{\vec{S}}_{i}\hat{\vec{S}}_{j}$ is investigated. It is shown that the quantum spin pair-correlator is equal to the correlator of classically evaluated vector field averaged over the initial conditions with respect to the gaussian measure. In the continious limit case the scaling estimations allow one to find one-point correlator that turns out to be $C(\vec{r}=0;t)\propto const \times t^{-6/7}$. All results are obtained by straightforward procedure without any assumptions of the phenomenological character.

cond-mat↗