1. Dynamics and its stability of Boltzmann-machine learning algorithm for gray scale image restoration J. Inoue (Hokkaido Univ.) and K. Tanaka (Tohoku Univ.) The 3 rd International Symposium on Slow Dynamics in Complex Systems in Sendai November 2003

2. Plan of this talk  Bayesian image restoration and hyper-parameter estimation  Boltzmann-machine learning algorithm for the hyper- parameter estimation  Dynamic behavior of the BML algorithm  Stability of the solution  Concluding remarks

3. Bayesian image restoration OriginalCorrupted We treat images and the degrading process as spin systems

4. Definitions of the model by spin systems Original Corrupted : Hyper-parameters (true value)

5. Bayesian approach and MPM estimation takes its minimum at [Inoue and Carlucci (2001)]

6. Maximization of the marginal likelihood via Boltzmann-machine learning algorithm takes its maximum aton average We evaluate the data-averaged BML algorithm at the mean-field level : [Inoue and Tanaka (2003)]

7. Dynamic behavior of the hyper-parameters are integrated numerically

8. Analysis of the stability Expand the BML equations around and check the sign of eigenvalues of the Hessian A The solution is asymptotically stable

9. True hyper-parameter dependence of the stability The solution of the BML algorithm is asymptotically stable as long as the solution is identical to the true value of the hyper-parameters (fixed)

10. Behavior of the BML algorithm around the solution Trajectories in the hyper-parameter space (around the solution)

11. Concluding remarks  We investigated dynamic behavior and its stability of the BML algorithm for gray scale image restoration  We derived the data-averaged BML equations  The solution is asymptotically stable as long as the solution is identical to the true value of the hyper- parameters  More details of the present study are available at http://chaosweb.complex.eng.hokudai.ac.jp/~j_inoue/ Send email : j_inoue@complex.eng.hokudai.ac.jp