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Conditional Random Fields, Planted Constraint Satisfaction and Entropy Concentration

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2013, RANDOM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8096))

Abstract

This paper studies a class of probabilistic models on graphs, where edge variables depend on incident node variables through a fixed probability kernel. The class includes planted constraint satisfaction problems (CSPs), as well as more general structures motivated by coding and community clustering problems. It is shown that under mild assumptions on the kernel and for sparse random graphs, the conditional entropy of the node variables given the edge variables concentrates around a deterministic threshold. This implies in particular the concentration of the number of solutions in a broad class of planted CSPs, the existence of a threshold function for the disassortative stochastic block model, and the proof of a conjecture on parity check codes. It also establishes new connections among coding, clustering and satisfiability.

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References

  1. Abbe, E., Montanari, A.: On the concentration of the number of solutions of random satisfiability formulas. Random Structures & Algorithms (2013) ISSN: 1098-2418, http://dx.doi.org/10.1002/rsa.20501 , doi:10.1002/rsa.20501

  2. Abbe, E., Montanari, A.: Conditional Random Fields, Planted Constraint Satisfaction, and Entropy Concentration. arXiv:1305.4274 [math.PR] (2013)

    Google Scholar 

  3. Achlioptas, D., Coja-Oghlan, A.: Algorithmic barriers from phase transitions. In: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, pp. 793–802. IEEE Computer Society, Washington, DC (2008)

    Chapter  Google Scholar 

  4. Krivelevich, M., Coja-Oghlan, A., Vilenchik, D.: Why almost all satisfiable k-cnf formulas are easy. In: Proceedings of the 13th International Conference on Analysis of Algorithms, pp. 89–102 (2007)

    Google Scholar 

  5. Achlioptas, D., Jia, H., Moore, C.: Hiding satisfying assignments: two are better than one. In: Proceedings of AAAI 2004, pp. 131–136 (2004)

    Google Scholar 

  6. Achlioptas, D., Kautz, H., Gomes, C.: Generating satisfiable problem instances

    Google Scholar 

  7. Achlioptas, D., Han Kim, J., Krivelevich, M., Tetali, P.: Two-coloring random hypergraphs. Random Structures and Algorithms 20(2), 249–259 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Altarelli, F., Monasson, R., Zamponi, F.: Can rare SAT formulas be easily recognized? On the efficiency of message passing algorithms for K-SAT at large clause-to-variable ratios. Computing Research Repository abs/cs/060 (2006)

    Google Scholar 

  9. Achlioptas, D., Naor, A., Peres, Y.: Rigorous Location of Phase Transitions in Hard Optimization Problems. Nature 435, 759–764 (2005)

    Article  Google Scholar 

  10. Bayati, M., Gamarnik, D., Tetali, P.: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In: 42nd Annual ACM Symposium on Theory of Computing, Cambridge, MA, pp. 105–114 (June 2010)

    Google Scholar 

  11. Barthel, W., Hartmann, A.K., Leone, M., Ricci-Tersenghi, F., Weigt, M., Zecchina, R.: Hiding solutions in random satisfiability problems: A statistical mechanics approach. Phys. Rev. Lett. 88, 188701 (2002)

    Article  Google Scholar 

  12. Berlekamp, E., McEliece, R.J., Van Tilborg, H.C.A.: On the inherent intractability of certain coding problems (corresp.). IEEE Transactions on Information Theory 24(3), 384–386 (1978)

    Article  MATH  Google Scholar 

  13. Coja-oghlan, A.: Graph partitioning via adaptive spectral techniques. Comb. Probab. Comput. 19(2), 227–284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dyer, M.E., Frieze, A.M.: The solution of some random np-hard problems in polynomial expected time. Journal of Algorithms 10(4), 451–489 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R., Rink, M.: Tight thresholds for cuckoo hashing via XORSAT. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 213–225. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84, 066106 (2011)

    Google Scholar 

  17. Dubois, O., Mandler, J.: The 3-XORSAT threshold. In: Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS 2002, pp. 769–778. IEEE Computer Society, Washington, DC (2002)

    Google Scholar 

  18. Daudé, H., Ravelomanana, V.: Random 2-XORSAT at the satisfiability threshold. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 12–23. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  19. Elias, P.: Coding for noisy channels. IRE Convention Record 4, 37–46 (1955)

    Google Scholar 

  20. Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111, 535 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Franz, S., Leone, M., Toninelli, F.L.: Replica bounds for diluted non-Poissonian spin systems. J. Phys. A 36, 10967 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Feige, U., Mossel, E., Vilenchik, D.: Complete convergence of message passing algorithms for some satisfiability problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX and RANDOM 2006. LNCS, vol. 4110, pp. 339–350. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  23. Fortunato, S.: Community detection in graphs. Physics Reports 486(3-5), 75–174 (2010)

    Article  MathSciNet  Google Scholar 

  24. Friedgut, E.: Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc. 12, 1017–1054 (1999); appendix by Bourgain, J.

    Google Scholar 

  25. Gallager, R.G.: Low-density parity-check codes. MIT Press, Cambridge (1963)

    Google Scholar 

  26. Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glasses. Commun. Math. Phys. 230, 71–79 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Goldenberg, A., Zheng, A.X., Fienberg, S.E., Airoldi, E.M.: A survey of statistical network models. Foundations and Trends® in Machine Learning 2(2), 129–233 (2010)

    Article  Google Scholar 

  28. Haanpää, H., Järvisalo, M., Kaski, P., Niemelä, I.: Hard satisfiable clause sets for benchmarking equivalence reasoning techniques (2005)

    Google Scholar 

  29. Jia, H., Moore, C., Strain, D.: Generating hard satisfiable formulas by hiding solutions deceptively. In: AAAI, pp. 384–389. AAAI Press (2005)

    Google Scholar 

  30. Kudekar, S., Macris, N.: Sharp bounds for optimal decoding of Low-Density Parity-Check codes. IEEE Trans. on Inform. Theory 55, 4635–4650 (2009)

    Article  MathSciNet  Google Scholar 

  31. Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011)

    Google Scholar 

  32. Krzakala, F., Zdeborová, L.: Hiding quiet solutions in random constraint satisfaction problems. Phys. Rev. Lett. 102, 238701 (2009)

    Article  Google Scholar 

  33. Lafferty, J.: Conditional random fields: Probabilistic models for segmenting and labeling sequence data, pp. 282–289. Morgan Kaufmann (2001)

    Google Scholar 

  34. Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D.A., Stemann, V.: Practical loss-resilient codes. In: 29th Annual ACM Symposium on Theory of Computing, pp. 150–159 (1997)

    Google Scholar 

  35. Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D.A.: Efficient erasure correcting codes. IEEE Trans. on Inform. Theory 47(2), 569–584 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mossel, E., Neeman, J., Sly, A.: Stochastic Block Models and Reconstruction. arXiv:1202.1499 [math.PR]

    Google Scholar 

  37. Montanari, A.: Tight bounds for LDPC and LDGM codes under MAP decoding. IEEE Trans. on Inform. Theory 51, 3221–3246 (2005)

    Article  MathSciNet  Google Scholar 

  38. Montanari, A.: Estimating random variables from random sparse observations. European Transactions on Telecommunications 19(4), 385–403 (2008)

    Article  Google Scholar 

  39. Montanari, A., Restrepo, R., Tetali, P.: Reconstruction and Clustering in Random Constraint Satisfaction Problems. CoRR abs/0904.2751 (2009)

    Google Scholar 

  40. Newman, M.E.J.: Communities, modules and large-scale structure in networks. Nature Physics 8(1), 25–31 (2011)

    Article  Google Scholar 

  41. Pittel, B., Sorkin, G.B.: The Satisfiability Threshold for k-XORSAT. arXiv:1212.1905 (2012)

    Google Scholar 

  42. Panchenko, D., Talagrand, M.: Bounds for diluted mean-field spin glass models. Prob. Theor. Rel. Fields 130, 319–336 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Raj Kumar, K., Pakzad, P., Salavati, A.H., Shokrollahi, A.: Phase transitions for mutual information. In: 2010 6th International Symposium on Turbo Codes and Iterative Information Processing (ISTC), pp. 137–141 (2010)

    Google Scholar 

  44. Richardson, T., Urbanke, R.: Modern Coding Theory. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  45. Zdeborová, L., Krzakala, F.: Quiet planting in the locked constraint satisfaction problems. SIAM Journal on Discrete Mathematics 25(2), 750–770 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Program in Applied and Computational Mathematics, Princeton University, USA

    Emmanuel Abbe

  2. Departments of Electrical Engineering and Statistics, Stanford University, USA

    Andrea Montanari

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  1. Emmanuel Abbe
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  2. Andrea Montanari
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Editor information

Editors and Affiliations

  1. University of California, Berkeley, CA, USA

    Prasad Raghavendra

  2. Dept. of computer Science and Engineering, Pennsylvania State University, University Park, PA, USA

    Sofya Raskhodnikova

  3. Institute of Computer Science, University of Kiel, 24118, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, 1227, Carouge, Switzerland

    José D. P. Rolim

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Abbe, E., Montanari, A. (2013). Conditional Random Fields, Planted Constraint Satisfaction and Entropy Concentration. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_24

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  • DOI: https://doi.org/10.1007/978-3-642-40328-6_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40327-9

  • Online ISBN: 978-3-642-40328-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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