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@ 15/7/2003 Tokyo Institute of Technology 1 Propagating beliefs in spin- glass models Yoshiyuki Kabashima Dept. of Compt. Intel. & Syst.

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Presentation on theme: "@ 15/7/2003 Tokyo Institute of Technology 1 Propagating beliefs in spin- glass models Yoshiyuki Kabashima Dept. of Compt. Intel. & Syst."— Presentation transcript:

1 Hayashibara@SMAPIP @ 15/7/2003 Tokyo Institute of Technology 1 Propagating beliefs in spin- glass models Yoshiyuki Kabashima Dept. of Compt. Intel. & Syst. Sci. Tokyo Institute of Technology

2 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology2 Background and Motivation Active research on belief propagation (BP) in information sciences (IS) Similarity to methods in physics TMM & Bethe approx. Difference in interest Physics ⇒ obtained solutions IS ⇒ dynamics of the algorithm may cause unexpected developments in both fields

3 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology3 Purpose and Results Purpose ： analyze dynamics of BP when employed in spin-glass models Results: Macro. dyn. of BP ⇒ RS solution Micro. stability of BP ⇔ AT condition

4 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology4 Outline SG model in Bayesian framework Belief propagation Macro. dyn. and RS solution Micro. stability and AT condition Summary

5 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology5 Spin-glass models SG models on a random (Bethe) lattice K-body interaction C-bonds/spin Randomly constructed for other aspects

6 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology6 SM and Bayesian Statistics Boltzmann dist. = Bayes formula Magnetization = Posterior average

7 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology7 Graph Expression Expression by a bipartite graph … … …

8 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology8 Belief Propagation Iterative inference by passing beliefs … … …

9 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology9 More Precisely ： Posterior average when is left out. ： Effective field when comes in. ： Estimator

10 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology10 Macro. Dyn. vs. RS Solution Distribution (histogram) of beliefs Known result for finite C: Tree approximation (resampling graph/update) Density evolution ⇒ RS solution Vicente, Saad, YK (2000)Richardson & Urbanke (2001)

11 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology11 Novel Result for Infinite C Central limit theorem for infinite C Evolution of average and variance Natural iteration of RS SP eqs.

12 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology12 Experimental Validation SK model （ N=1000,J=1,T=0.5 ） (AT stable) (AT unstable)

13 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology13 Microscopic Instability Possible microscopic instability while BP seems to macroscopically converge Stability analysis of the fixed point

14 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology14 Evolution of Perturbation Dist. of Perturb. Perturbation Evolution is attractive ⇔ the fixed point(=RS solution) is stable

15 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology15 Pictorial Expression What is performed? … … … … … … … … …

16 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology16 Meaning of P. Evolution Link to known results for infinite C Central limit theorem ： P. Evolution → Update of average & variance :Gaussian dist.

17 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology17 Meaning of P. Evolution Critical conditions for growth of fluctuation For K=2 (SK model) Average → T f : Para-Ferro transition Variance → T AT : AT condition :Average :Variance

18 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology18 Analysis for finite C Is P. evolution equivalent to AT analysis even for finite C? AT analysis for finite C is complicated. But, P. evolution is (numerically) possible. K=2 (Wong-Sherrington model) Paramagnetic solution Average → T f Variance → T AT Known result Klein et al (1979) Mezard & Parisi (1987)

19 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology19 Analysis for Finite C Ferromagnetic solution Numerical evaluation of T pevol : New result! N=2000, K=2, C=4, 20000MCS/Spin  : T pevol in Ferro phase

20 Hayashibara&SMAPIP @15/7/2003 Tokyo Institute of Technology20 Summary Close relationship between BP and the replica analysis Macro. dyn. ⇒ RS solution Micro. stability ⇔ AT condition This correspondence may be useful for AT analysis for SG models of finite connectivity. Application to CDMA multiuser detection (Kabashima, to appear in J. Phys. A, 2003)

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