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.

[1]  M. Mézard,et al.  Random K-satisfiability problem: from an analytic solution to an efficient algorithm. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  D. Saad,et al.  Statistical mechanics of error-correcting codes , 1999 .

[3]  Yoshiyuki Kabashima.  Replicated Bethe Free Energy: A Variational Principle behind Survey Propagation , 2005 .

[4]  M. Mézard,et al.  A ferromagnet with a glass transition , 2001, cond-mat/0103026.

[5]  Michael I. Jordan,et al.  Advances in Neural Information Processing Systems 30 , 1995 .

[6]  B Wemmenhove,et al.  Survey propagation at finite temperature: application to a Sourlas code as a toy model , 2006 .

[7]  Ruján.  Finite temperature error-correcting codes. , 1993, Physical review letters.

[8]  Kwon,et al.  Ising spin glass at zero temperature on the Bethe lattice. , 1988, Physical review. B, Condensed matter.

[9]  Andrea Montanari,et al.  Instability of one-step replica-symmetry-broken phase in satisfiability problems , 2003, ArXiv.

[10]  D. Saad,et al.  Error-correcting codes that nearly saturate Shannon's bound , 1999, cond-mat/9906011.

[11]  N. Sourlas.  Spin Glasses, Error-Correcting Codes and Finite-Temperature Decoding , 1994 .

[12]  F. Krzakala,et al.  Spin glass models with ferromagnetically biased couplings on the Bethe lattice: analytic solutions and numerical simulations , 2004, cond-mat/0403053.

[13]  Y. Kabashima.  A CDMA multiuser detection algorithm on the basis of belief propagation , 2003 .

[14]  Ido Kanter,et al.  Finite-size effects and error-free communication in Gaussian channels , 2000 .

[15]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[16]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[17]  Nicolas Sourlas,et al.  Spin-glass models as error-correcting codes , 1989, Nature.

[18]  M. Mézard,et al.  The Cavity Method at Zero Temperature , 2002, cond-mat/0207121.

[19]  Kanter,et al.  Cascading parity-check error-correcting codes , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

[21]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[22]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[23]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .

[24]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[25]  Yoshiyuki Kabashima,et al.  Belief propagation vs. TAP for decoding corrupted messages , 1998 .

[26]  Y. Kabashima.  Propagating beliefs in spin-glass models , 2002, cond-mat/0211500.