A Quantum-Statistical-Mechanical Extension of Gaussian Mixture Model

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 IW-SMI2007, Kyoto

2. Contents • Introduction • Conventional Gaussian Mixture Model • Quantum Mechanical Extension of Gaussian Mixture Model • Quantum Belief Propagation • Concluding Remarks IW-SMI2007, Kyoto

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

4. 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? IW-SMI2007, Kyoto

5. Contents • Introduction • Conventional Gaussian Mixture Model • Quantum Mechanical Extension of Gaussian Mixture Model • Quantum Belief Propagation • Concluding Remarks IW-SMI2007, Kyoto

6. Prior of Gauss Mixture Model One of three labels 1,2 and 3 is assigned to each node. 3 labels Label xi is generated randomly and independently of each node. Histogram 2 3 1 xi=2 xi=3 xi=1 IW-SMI2007, Kyoto

7. Date Generating Process Data yi are generated randomly and independently of each node. xi=1 xi=2 xi=3 IW-SMI2007, Kyoto

8. Gauss Mixture Models Prior Probability Data Generating Process Marginal Likelihood for Hyperparameters m,s and a IW-SMI2007, Kyoto

9. Conventional Gauss Mixture Models a, m, s r(yi) Labels IW-SMI2007, Kyoto

10. Contents • Introduction • Conventional Gaussian Mixture Model • Quantum Mechanical Extension of Gaussian Mixture Model • Quantum Belief Propagation • Concluding Remarks IW-SMI2007, Kyoto

11. Quantum Gauss Mixture Models IW-SMI2007, Kyoto

12. Quantum Gauss Mixture Models Quantum Representation IW-SMI2007, Kyoto

13. Quantum Gauss Mixture Models IW-SMI2007, Kyoto

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

15. Quantum Gauss Mixture Models a, m, s r(yi) IW-SMI2007, Kyoto

16. Image Segmentation g = 0.4 g = 0.2 Conventional Gauss Mixture Model Original Image Quantum Gauss Mixture Model Histogram 0 255 0 255 0 255 IW-SMI2007, Kyoto

17. Image Segmentation g = 1.0 g = 0.5 Conventional Gauss Mixture Model Original Image Quantum Gauss Mixture Model Histogram 0 255 0 255 0 255 IW-SMI2007, Kyoto

18. Contents • Introduction • Conventional Gaussian Mixture Model • Quantum Mechanical Extension of Gaussian Mixture Model • Quantum Belief Propagation • Concluding Remarks IW-SMI2007, Kyoto

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

20. Image Segmentation Gauss Mixture Model Gauss Mixture Model and Potts Model Original Image Histogram Belief Propagation IW-SMI2007, Kyoto

21. j i Density Matrix and Reduced Density Matrix Reduced Density Matrix Reducibility Condition IW-SMI2007, Kyoto

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

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

24. Contents • Introduction • Conventional Gaussian Mixture Model • Quantum Mechanical Extension of Gaussian Mixture Model • Quantum Belief Propagation • Concluding Remarks IW-SMI2007, Kyoto

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