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Data clustering with quantum mechanics. (English) Zbl 1367.81009
Summary: Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.
81-08 Computational methods for problems pertaining to quantum theory
91C20 Clustering in the social and behavioral sciences
62H30 Classification and discrimination; cluster analysis (statistical aspects)
Full Text: DOI
[1] Lloyd, S.P.; Least squares quantization in PCM; IEEE Trans. Inform. Theory: 1982; Volume 28 ,129-137. · Zbl 0504.94015
[2] Huang, M.; Yu, L.; Chen, Y.; Improved K-Means Clustering Center Selecting Algorithm; Information Engineering and Applications, Proceedings of the International Conference on Information Engineering and Applications (IEA 2011), Chongqing, China, 21-24 October 2011: London, UK 2012; ,373-379.
[3] Girisan, E.; Thomas, N.A.; An Efficient Cluster Centroid Initialization Method for K-Means Clustering; Autom. Auton. Syst.: 2012; Volume 4 ,1.
[4] Dunn, J.C.; A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters; J. Cybernet.: 1973; Volume 3 ,32-57. · Zbl 0291.68033
[5] Brillouin, L.; ; Science and Information Theory: Dover, UK 1956; . · Zbl 0071.13104
[6] Georgescu-Roegencholas, N.; ; The Entropy Law and the Economic Process: Cambridge, MA, USA 1971; .
[7] Chen, J.; ; The Physical Foundation of Economics—An Analytical Thermodynamic Theory: London, UK 2005; . · Zbl 1140.91009
[8] Lin, S.K.; Diversity and Entropy; Entropy: 1999; Volume 1 ,101-104.
[9] Buhmann, J.M.; Hofmann, T.; A Maximum Entropy Approach to Pairwise Data Clustering; Conference A: Computer Vision & Image Processing, Proceedings of the 12th IAPR International Conference on Pattern Recognition, Jerusalem, Israel, 9-13 October 1994: Hebrew University, Jerusalem, Israel 1994; Volume Volume II ,207-212.
[10] Hofmann, T.; Buhmann, J.M.; Pairwise Data Clustering by Deterministic Annealing; IEEE Trans. Pattern Anal. Mach. Intell.: 1997; Volume 19 ,1-14.
[11] Zhu, S.; Ji, X.; Xu, W.; Gong, Y.; Multi-labelled Classification Using Maximum Entropy Method; Proceedings of the 28th Annual International ACM SIGIR Conference on Research and Development in Information (SIGIR’05): ; ,207-212.
[12] Coifman, R.R.; Lafon, S.; Lee, A.B.; Maggioni, M.; Nadler, B.; Warner, F.; Zucker, S.; Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data: Diffusion Maps; Proc. Natl. Acad. Sci. USA: 2005; Volume 102 ,7426-7431. · Zbl 1405.42043
[13] Meila, M.; Shi, J.; Learning Segmentation by Random Walks; Neural Inform. Process. Syst.: 2001; Volume 13 ,873-879.
[14] ; Applied Probability and Queues: New York, NY, USA 2003; ,3-38.
[15] Hammond, B.L.; Lester, W.A.; Reynolds, P.J.; EBSCOhost; Monte Carlo Methods in Ab Initio Quantum Chemistry: Singapore 1994; ,287-304.
[16] Lüchow, A.; Quantum Monte Carlo methods; Wiley Interdiscip. Rev. Comput. Mol. Sci.: 2011; Volume 1 ,388-402.
[17] Park, J.L.; The concept of transition in quantum mechanics; Found. Phys.: 1970; Volume 1 ,23-33.
[18] Louck, J.D.; Doubly stochastic matrices in quantum mechanics; Found. Phys.: 1997; Volume 27 ,1085-1104.
[19] Lafon, S.; Lee, A.B.; Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization; IEEE Trans. Pattern Anal. Mach. Intell.: 2006; Volume 28 ,1393-1403.
[20] Nadler, B.; Lafon, S.; Coifman, R.R.; Kevrekidis, I.G.; Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators; Advances in Neural Information Processing Systems 18: Cambridge, MA, USA 2005; ,955-962.
[21] Bogolyubov, N.; Sankovich, D.P.; N. N. Bogolyubov and Statistical Mechanics; Russ. Math. Surv.: 1994; Volume 49 ,19. · Zbl 0852.60103
[22] Brics, M.; Kaupuzs, J.; Mahnke, R.; How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?; Condens. Matter Phys.: 2013; Volume 16 ,13002.
[23] Lüchow, A.; Scott, T.C.; Nodal structure of Schrüdinger wavefunction: General results and specific models; J. Phys. B: At. Mol. Opt. Phys.: 2007; Volume 40 ,851.
[24] Lüchow, A.; Petz, R.; Scott, T.C.; Direct optimization of nodal hypersurfaces in approximate wave functions; J. Chem. Phys.: 2007; Volume 126 ,144110.
[25] Cheng, D.; Vempala, S.; Kannan, R.; Wang, G.; A Divide-and-merge Methodology for Clustering; Proceedings of the Twenty-fourth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS ’05): New York, NY, USA 2005; ,196-205.
[26] Golub, G.H.; Van Loan, C.F.; Matrix Computations; Johns Hopkins Studies in the Mathematical Sciences: Baltimore, MD, USA 1996; . · Zbl 0865.65009
[27] Horn, D.; Gottlieb, A.; Algorithm for data clustering in pattern recognition problems based on quantum mechanics; Phys. Rev. Lett.: 2002; Volume 88 ,18702.
[28] COMPACT Software Package; ; .
[29] Jones, M.; Marron, J.; Sheather, S.; A brief survey of bandwidth selection for density estimation; J. Am. Stat. Assoc.: 1996; Volume 91 ,401-407. · Zbl 0873.62040
[30] Brand, M.; Fast low-rank modifications of the thin singular value decomposition; Linear Algebra Appl.: 2006; Volume 415 ,20-30. · Zbl 1088.65037
[31] Sleipjen, G.L.G.; van der Vorst, H.A.; A Jacobi-Davidson iteration method for linear eigenvalue problems; SIAM J. Matrix Anal. Appl.: 1996; Volume 17 ,401-425. · Zbl 0860.65023
[32] Steffen, B.; Subspace Methods for Large Sparse Interior Eigenvalue Problems; Int. J. Differ. Equ. Appl.: 2001; Volume 3 ,339-351. · Zbl 1045.65033
[33] Voss, H.; A Jacobi-Davidson Method for Nonlinear Eigenproblems; Proceedings of the 4th International Conference on Computational Science (ICCS 2004): Berlin/Heidelberg, Germany 2004; ,34-41. · Zbl 1080.65535
[34] Stathopoulos, A.; PReconditioned Iterative MultiMethod Eigensolver; ; . · Zbl 1364.65087
[35] Stathopoulos, A.; McCombs, J.R.; PRIMME: Preconditioned Iterative Multimethod Eigensolver—Methods and Software Description; ACM Trans. Math. Softw.: 2010; Volume 37 ,1-29. · Zbl 1364.65087
[36] Larsen, R.M.; Computing the SVD for Large and Sparse Matrices, SCCM & SOI-MDI; 2000; .
[37] Chen, W.Y.; Song, Y.; Bai, H.; Lin, C.J.; Chang, E.Y.; Parallel Spectral Clustering in Distributed Systems; IEEE Trans. Pattern Anal. Mach. Intell.: 2011; Volume 33 ,568-586.
[38] Zhang, B.; Estrada, T.; Cicotti, P.; Taufer, M.; On Efficiently Capturing Scientific Properties in Distributed Big Data without Moving the Data: A Case Study in Distributed Structural Biology using MapReduce; Proceedings of the 16th IEEE International Conferences on Computational Science and Engineering (CSE): ; .
[39] Ripley, B.; ; Pattern Recognition and Neural Networks: Cambridge, UK 1996; . · Zbl 0853.62046
[40] Ripley, B.; CRAB DATA, 1996; ; .
[41] Jaccard, P.; Etude comparative de la distribution florale dans une portion des Alpes et des Jura; Bull. Soc. Vaud. Sci. Nat.: 1901; Volume 37 ,547-579.
[42] Hearst, M.; Untangling Text Data Mining, 1999; ; .
[43] Wang, S.; ; Thematic Clustering and the Dual Representations of Text Objects: ; .
[44] Wang, S.; Dignan, T.G.; Thematic Clustering; U.S. Patent: 2014; .
[45] Strehl, A.; strehl.com; ; .
[46] Ben-Hur, A.; Horn, D.; Siegelmann, H.T.; Vapnik, V.; Support Vector Clustering; J. Mach. Learn. Res.: 2002; Volume 2 ,125-137. · Zbl 1002.68598
[47] The Method of Quantum Clustering; Advances in Neural Information Processes: Cambridge, MA, USA 2002; .
[48] Draper, N.; Smith, H.; ; Applied Regression Analysis: New York, NY, USA 1981; . · Zbl 0548.62046
[49] Miller, F.R.; Neill, J.W.; Sherfey, B.W.; Maximin Clusters from near-replicate Regression of Fit Tests; Ann. Stat.: 1998; Volume 26 ,1411-1433. · Zbl 0932.62075
[50] ; Expoplanet.eu—Extrasolar Planets Encyclopedia: ; .
[51] Yaqoob, T.; ; Exoplanets and Alien Solar Systems: Baltimore, MD, USA 2011; .
[52] Fertik, M.; Scott, T.; Dignan, T.; Identifying Information Related to a Particular Entity from Electronic Sources, Using Dimensional Reduction and Quantum Clustering; U.S. Patent: 2014; .
[53] Bekkerman, R.; McCallum, A.; Disambiguating Web Appearances of People in a Social Network; 2005; .
[54] Zeimpekis, D.; Gallopoulos, E.; TMG: A MATLAB Toolbox for Generating Term-Document Matrices from Text Collections; 2005; .
[55] Ding, J.; Zhou, A.; Eigenvalues of rank-one updated matrices with some applications; Appl. Math. Lett.: 2007; Volume 20 ,1223-1226. · Zbl 1139.15003
[56] Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G.A.; ; Gaussian-09 Revision E.01: Wallingford, CT, USA 2009; .
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