1. September 2007 IW-SMI2007, Kyoto 1 A Quantum-Statistical-Mechanical Extension of Gaussian Mixture Model Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Sendai, Japan http://www.smapip.is.tohoku.ac.jp/~kazu/ In collaboration with Koji Tsuda Max Planck Institute for Biological Cybernetics, Germany

2. September 2007IW-SMI2007, Kyoto2 Contents 1.Introduction 2.Conventional Gaussian Mixture Model 3.Quantum Mechanical Extension of Gaussian Mixture Model 4.Quantum Belief Propagation 5.Concluding Remarks

3. September 2007IW-SMI2007, Kyoto3 Information Processing by using Quantum Statistical-Mechanics Quantum Annealing in Optimizations Quantum Error Correcting Codes etc... Massive Information Processing by means of Density Matrix

4. September 2007IW-SMI2007, Kyoto4 Motivations How can we construct the quantum Gaussian mixture model? How can we construct a data- classification algorithm by using the quantum Gaussian mixture model?

5. September 2007IW-SMI2007, Kyoto5 Contents 1.Introduction 2.Conventional Gaussian Mixture Model 3.Quantum Mechanical Extension of Gaussian Mixture Model 4.Quantum Belief Propagation 5.Concluding Remarks

6. September 2007IW-SMI2007, Kyoto6 Prior of Gauss Mixture Model 1 32 Histogram Label x i is generated randomly and independently of each node. 3 labels x i =1 x i =2x i =3 One of three labels 1,2 and 3 is assigned to each node.

7. September 2007IW-SMI2007, Kyoto7 Date Generating Process Data y i are generated randomly and independently of each node. x i =1 x i =2 x i =3

8. September 2007IW-SMI2007, Kyoto8 Gauss Mixture Models Prior Probability Data Generating Process Marginal Likelihood for Hyperparameters ,  and 

9. September 2007IW-SMI2007, Kyoto9 Conventional Gauss Mixture Models , ,  (yi)(yi) Labels

10. September 2007IW-SMI2007, Kyoto10 Contents 1.Introduction 2.Conventional Gaussian Mixture Model 3.Quantum Mechanical Extension of Gaussian Mixture Model 4.Quantum Belief Propagation 5.Concluding Remarks

11. September 2007IW-SMI2007, Kyoto11 Quantum Gauss Mixture Models

12. September 2007IW-SMI2007, Kyoto12 Quantum Gauss Mixture Models Quantum Representation

13. September 2007IW-SMI2007, Kyoto13 Quantum Gauss Mixture Models

14. September 2007IW-SMI2007, Kyoto14 Linear Response Formulas Maxmum Likelihood Estimation in Quantum Gauss Mixture Model

15. September 2007IW-SMI2007, Kyoto15 Quantum Gauss Mixture Models , ,  (yi)(yi)

16. September 2007IW-SMI2007, Kyoto16 Image Segmentation Original Image Histogram Conventional Gauss Mixture Model Quantum Gauss Mixture Model  = 0.2  = 0.4 0 255 0 0

17. September 2007IW-SMI2007, Kyoto17 Image Segmentation Original Image Histogram Conventional Gauss Mixture Model Quantum Gauss Mixture Model  = 0.5  = 1.0 02550 0

18. September 2007IW-SMI2007, Kyoto18 Contents 1.Introduction 2.Conventional Gaussian Mixture Model 3.Quantum Mechanical Extension of Gaussian Mixture Model 4.Quantum Belief Propagation 5.Concluding Remarks

19. September 2007 IW-SMI2007, Kyoto 19 Image Segmentation by Combining Gauss Mixture Model with Potts Model Belief Propagation = = > Potts Model 4 labels

20. September 2007IW-SMI2007, Kyoto20 Image Segmentation Original Image Histogram Gauss Mixture Model Gauss Mixture Model and Potts Model BeliefPropagation

21. September 2007IW-SMI2007, Kyoto21 Density Matrix and Reduced Density Matrix Reduced Density Matrix Reducibility Condition j i

22. September 2007IW-SMI2007, Kyoto22 Reduced Density Matrix and Effective Fields i j i All effective field are matrices

23. September 2007IW-SMI2007, Kyoto23 Belief Propagation for Quantum Statistical Systems Propagating Rule of Effective Fields j i Output

24. September 2007IW-SMI2007, Kyoto24 Contents 1.Introduction 2.Conventional Gaussian Mixture Model 3.Quantum Mechanical Extension of Gaussian Mixture Model 4.Quantum Belief Propagation 5.Concluding Remarks

25. September 2007IW-SMI2007, Kyoto25 Summary An Extension to Quantum Statistical Mechanical Gaussian Mixture Model Practical Algorithm  Linear Response Formula Application of Potts Model and Quantum Belief Propagation Applications to Data Mining Extension to Quantum Deterministic Annealing Future Problem