近年來,有許多以量子力學概念為基礎的演化式計算方法被提出並應用在解決組合最佳化問題上。在這些方法中,有些是特別針對資料分群問題求解的方法。然而,利用這些方法卻無法求得令人滿意的分群結果。因此,本研究提出一個新的量子演化式計算方法來解決資料分群問題,稱為「量子啟發式演化分群演算法」。本研究所提出的方法在以傳統量子演化法為基礎的架構上,加入分群解的概念與 k-means 機制,同時提出有效的修復機制來提升量子啟發式演化分群演算法的效能。為了評估本論文提出的方法之效能,特別針對真實環境資料集進行分群並比較量子啟發式演化分群演算法與傳統量子演化法以及 genetic k-means algorithm 的資料分群結果。從實驗結果顯示,本論文提出的方法是有效且可行的,同時也是非常具有潛力的方法。

In recent years, a lot of evolutionary computation methods have been proposed to solve the combinatorial optimization problem based on the concepts of quantum mechanics. Although some of them are purposely presented for solving the data clustering problem, they are all far from optimal quality-wise. As such, this thesis proposes a new method, called quantum-inspired evolutionary clustering algorithm (QECA), to address the data clustering problem. The proposed method adds not only the concepts of clustering and the k-means to the traditional quantum-inspired evolutionary algorithm to make it work for clustering but also an effective repair operator to improve the performance of the proposed method. Experimental results on real world data show that the proposed method provides a promising result than those obtained by QEA and genetic k-means algorithm.

論文審定書 i
Acknowledgments iii
摘要 v
Abstract vi
List of Figures x
List of Tables xii
Chapter 1 Introduction 1
1.1 Background 1
1.2 Motivation 3
1.3 Contributions 3
1.4 Organization 3
Chapter 2 Related Works 5
2.1 Clustering Problem and Related Algorithms 5
2.1.1 Taxonomy of Clustering Problem 5
2.1.2 Definition of Clustering Problem 7
2.1.3 Basis of Clustering Activity 8
2.1.4 Similarity of Clustering Problem 9
2.1.5 k-means algorithm 10
2.1.6 Genetic k-means Algorithm (GKA) 12
2.1.6.1 Selection Operation 13
2.1.6.2 Mutation Operation 14
2.1.6.3 k-means Operation 15
2.2 Quantum Computing 16
2.2.1 Background of Quantum Computing 16
2.2.2 Principles of Quantum Computation 19
2.2.3 Quantum-inspired Computing 19
2.3 Quantum-inspired Evolutionary Algorithm 22
2.3.1 Quantum-inspired Evolutionary Algorithm (QEA) 22
2.3.1.1 Representation of QEA 22
2.3.1.2 Overall Design of QEA 23
2.3.2 QEA for Clustering 27
2.3.2.1 Initialization 27
2.3.2.2 Representation and Fitness Function 28
2.3.2.3 Make and Repair Operation 28
2.3.2.4 Evaluate and Update Operation 29
Chapter 3 The Proposed Method 31
3.1 Concepts and Representation 31
3.1.1 Concepts 31
3.1.2 Representation 31
3.2 Structure of the Proposed Method 34
3.3 Algorithm of the Proposed Method 37
3.3.1 Initialization Operation 37
3.3.2 Observation Operation 37
3.3.3 Repair Operation 38
3.3.4 Clustering Operation 41
3.3.5 k-means Operation 42
3.3.6 Evaluation Operation 43
3.3.7 Update Operation 43
3.3.8 Store Best Operations 47
3.3.9 Migration Operation 48
3.4 Difference between QECA and QEA 48
3.5 Application Example 50
Chapter 4 Experiments 55
4.1 Environment, Datasets and Parameter Settings 55
4.1.1 Environment 55
4.1.2 Datasets and Parameter Settings 56
4.2 Simulation Results 58
4.3 Analysis 61
4.3.1 Rotation Angle 61
4.3.1.1 Convergence of Q-bits 61
4.3.1.2 The Value of Rotation Angle 64
4.3.2 Trends of Optimum Solution 67
4.3.3 Population Size 72
Chapter 5 Conclusions and Future Works 76
5.1 Conclusions 76
5.2 Future Works 77
Bibliography 78
Appendix A Rotation Angle and Corresponding SSE Value 84

[1] A. K. Jain, M. N. Murty, and P. J. Flynn, “Data clustering: a review,” ACM computing surveys (CSUR), vol. 31, no. 3, pp. 264–323, 1999.
[2] A. K. Jain and R. C. Dubes, Algorithms for clustering data. Prentice-Hall, Inc., 1988.
[3] J. A. Hartigan, Clustering algorithms. New York, USA: John Wiley & Sons, Inc., 1975.
[4] M. R. Anderberg, Cluster analysis for applications. Academic Press, 1973.
[5] W. J. Welch, “Algorithmic complexity: Three NP-hard problems in computational statistics,” Journal of Statistical Computation and Simulation, vol. 15, no. 1, pp. 17–25, 1982.
[6] W. Xu, X. Liu, and Y. Gong, “Document clustering based on non-negative matrix factorization,” in Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval, pp. 267–273, The Association for Computing Machinery, 2003.
[7] W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: A literature survey,” ACM Computing Surveys (CSUR), vol. 35, no. 4, pp. 399–458, 2003.
[8] G. Getz, H. Gal, I. Kela, D. A. Notterman, and E. Domany, “Coupled two-way clustering analysis of breast cancer and colon cancer gene expression data,” Bioinformatics, vol. 19, no. 9, pp. 1079–1089, 2003.
[9] P. Ferragina and A. Gulli, “A personalized search engine based on web-snippet hierarchical clustering,” in Special interest tracks and posters of the 14th international conference on World Wide Web, pp. 801–810, 2005.
[10] U. Maulik and S. Bandyopadhyay, “Genetic algorithm-based clustering technique,” Pattern Recognition, vol. 33, no. 9, pp. 1455–1465, 2000.
[11] X. Cui, T. E. Potok, and P. Palathingal, “Document clustering using particle swarm optimization,” in Proceedings of the IEEE Swarm Intelligence Symposium, pp. 185–191, Institute of Electrical and Electronics Engineers, 2005.
[12] K. Krishna and M. N. Murty, “Genetic k-means algorithm,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 29, pp. 433–439, 1999.
[13] D. W. V. D. Merwe and A. P. Engelbrecht, “Data clustering using particle swarm optimization,” in Proceedings of the 2003 IEEE Congress on Evolutionary Computation, vol. 1, pp. 215–220, 2003.
[14] R. Kicinger, T. Arciszewski, and K. D. Jong, “Evolutionary computation and structural design: A survey of the state-of-the-art,” Computers & Structures, vol. 83, no. 23, pp. 1943–1978, 2005.
[15] T. Bäck, Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms. Oxford University Press, 1996.
[16] T. Bäck and H.-P. Schwefel, “An overview of evolutionary algorithms for parameter optimization,” Evolutionary Computation, vol. 1, no. 1, pp. 1–23, 1993.
[17] W. M. Spears, K. A. D. Jong, T. Bäck, D. B. Fogel, and H. D. Garis, “An overview of evolutionary computation,” in Proceedings of the 1993 European Conference on Machine Learning, pp. 442–459, Springer, 1993.
[18] K.-H. Han and J.-H. Kim, “Quantum-inspired evolutionary algorithm for a class of combinatorial optimization,” IEEE Transactions on Evolutionary Computation, vol. 6, no. 6, pp. 580–593, 2002.
[19] J. H. Holland, Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, 1975.
[20] L. J. Fogel, A. J. Owens, and M. J. Walsh, Artificial intelligence through simulated evolution. John Wiley & Sons, Inc., 1966.
[21] H.-P. P. Schwefel, Evolution and optimum seeking: the sixth generation. John Wiley & Sons, Inc., 1993.
[22] J. R. Koza, Genetic programming: on the programming of computers by means of natural selection, vol. 1. MIT press, 1992.
[23] P. J. Bentley, T. G. W. Gordon, J. Kim, and S. Kumar, “New trends in evolutionary computation,” in Proceedings of the 2001 IEEE Congress on Evolutionary Computation, vol. 1, pp. 162–169, Institute of Electrical and Electronics Engineers, 2001.
[24] W. Zhou, C. Zhou, Y. Huang, and Y. Wang, “Analysis of gene expression data: application of quantum-inspired evolutionary algorithm to minimum sum-of-squares clustering,” in Proceedings of the 10th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, pp. 383–391, Springer, 2005.
[25] D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Transactions on Evolutionary Computation, vol. 1, no. 1, pp. 67–82, 1997.
[26] C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: Overview and conceptual comparison,” ACM Computing Surveys (CSUR), vol. 35, pp. 268–308, September 2003.
[27] R. Xu and D. Wunsch, “Survey of clustering algorithms,” IEEE Transactions on Neural Networks, vol. 16, no. 3, pp. 645–678, 2005.
[28] J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297, University of California Press, 1967.
[29] J. Xiao, Y. Yan, J. Zhang, and Y. Tang, “A quantum-inspired genetic algorithm for k-means clustering,” Expert Systems with Applications, vol. 37, no. 7, pp. 4966–4973, 2010.
[30] D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning, vol. 412. Addison-Wesley Professional, 1989.
[31] A. Narayanan, “Quantum computing for beginners,” in Proceedings of the 1999 IEEE Congress on Evolutionary Computation, vol. 3, pp. 2231–2238, 1999.
[32] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge university press, 2010.
[33] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Physical Review Letters, vol. 70, pp. 1895–1899, March 1993.
[34] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” Journal of Cryptology, vol. 5, no. 1, pp. 3–28, 1992.
[35] A. M. Turing, “On computable numbers, with an application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society, vol. 2, no. 42, pp. 230–265, 1936.
[36] R. Landauer, “Irreversibility and heat generation in the computing process,” IBM Journal of Research and Development, vol. 5, pp. 183–191, July 1961.
[37] C. H. Bennett, “Logical reversibility of computation,” IBM journal of Research and Development, vol. 17, pp. 525–532, November 1973.
[38] P. Benioff, “The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines,” Journal of Statistical Physics, vol. 22, no. 5, pp. 563–591, 1980.
[39] P. Benioff, “Quantum mechanical models of Turing machines that dissipate no energy,” Physical Review Letters, vol. 48, no. 23, pp. 1581–1585, 1982.
[40] R. P. Feynman, “Simulating physics with computers,” International Journal of Theoretical Physics, vol. 21, no. 6, pp. 467–488, 1982.
[41] C. H. Bennett and R. Landauer, “The fundamental physical limits of computation,” Scientific American, vol. 253, no. 1, pp. 48–56, 1985.
[42] D. Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proceedings of the Royal Society of London, vol. 400, no. 1818, pp. 97–117, 1985.
[43] D. Deutsch, “Quantum computational networks,” Proceedings of the Royal Society of London, vol. 425, no. 1868, pp. 73–90, 1989.
[44] D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proceedings of the Royal Society of London, vol. 439, no. 1907, pp. 553–558, 1992.
[45] P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134, Institute of Electrical and Electronics Engineers, 1994.
[46] D. R. Simon, “On the power of quantum computation,” Society for Industrial and Applied Mathematics Journal on Computing (SICOMP), vol. 26, no. 5, pp. 1474–1483, 1994.
[47] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 212–219, The Association for Computing Machinery, 1996.
[48] M. Moore and A. Narayanan, “Quantum-inspired computing,” 1995.
[49] T. Hey, “Quantum computing: an introduction,” Computing & Control Engineering Journal, vol. 10, no. 3, pp. 105–112, 1999.
[50] A. O. Pittenger, An Introduction to Quantum Computing Algorithms, vol. 19. Birkhäuser, 1999.
[51] J. Preskill, “Lecture notes for physics 229: Quantum information and computation,” 1998.
[52] A. Narayanan and M. Moore, “Quantum-inspired genetic algorithms,” in Proceedings of the 1996 IEEE International Conference on Evolutionary Computation, pp. 61–66, Institute of Electrical and Electronics Engineers, IEEE Press, 1996.
[53] K.-H. Han and J.-H. Kim, “Quantum-inspired evolutionary algorithms with a new termination criterion, Hε gate, and two-phase scheme,” IEEE Transactions on Evolutionary Computation, vol. 8, no. 2, pp. 156–169, 2004.
[54] J. Sun, W. Xu, and B. Feng, “A global search strategy of quantum-behaved particle swarm optimization,” in Proceedings of the 2004 IEEE Conference on Cybernetics and Intelligent Systems, vol. 1, pp. 111–116, Institute of Electrical and Electronics Engineers, 2004.
[55] Y. Li, Y.-N. Zhang, R.-C. Zhao, and L.-C. Jiao, “An edge detector based on parallel quantum-inspired evolutionary algorithm,” in Proceedings of the 2004 International Conference on Machine Learning and Cybernetics, vol. 7, pp. 4062–4066, Institute of Electrical and Electronics Engineers, August 2004.
[56] Y. Li, Y.-N. Zhang, R.-C. Zhao, and L.-C. Jiao, “The immune quantum-inspired evolutionary algorithm,” in Proceedings of the 2014 IEEE International Conference on Systems, Man, and Cybernetics, vol. 4, pp. 3301–3305, Institute of Electrical and Electronics Engineers, 2004.
[57] J. Sun, W. Xu, and B. Ye, “Quantum-behaved particle swarm optimization clustering algorithm,” in Advanced Data Mining and Applications, vol. 4093, Springer, 2006.
[58] W. Chen, J. Sun, Y. Ding, W. Fang, and W. Xu, “Clustering of gene expression data with quantum-behaved particle swarm optimization,” in New Frontiers in Applied Artificial Intelligence, vol. 5027 of Lecture Notes in Computer Science, pp. 388–396, Springer, 2008.
[59] J. Xiao, Y. Yan, Y. Lin, L. Yuan, and J. Zhang, “A quantum-inspired genetic algorithm for data clustering,” in Proceedings of the 2008 IEEE Congress on Evolutionary Computation, pp. 1513–1519, Institute of Electrical and Electronics Engineers, 2008.
[60] S. Li and H. Li, “Quantum immune evolutionary algorithm with application in clustering,” Journal of Information & Computational Science, vol. 9, no. 5, pp. 1167–1174, 2012.
[61] K. Bache and M. Lichman, “UCI machine learning repository,” 2013.
[62] K. L. Priddy and P. E. Keller, Artificial neural networks: an introduction, vol. 68. SPIE Press, 2005.