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Natural Computing

, Volume 18, Issue 4, pp 907–911 | Cite as

Data clustering based on quantum synchronization

  • Aladin CrnkićEmail author
  • Vladimir Jaćimović
Article

Abstract

There exists a specific class of methods for data clustering problem inspired by synchronization of coupled oscillators. This approach requires an extension of the classical Kuramoto model to higher dimensions. In this paper, we propose a novel method based on so-called non-Abelian Kuramoto models. These models provide a natural extension of the classical Kuramoto model to the case of abstract particles (called Kuramoto–Lohe oscillators) evolving on matrix Lie groups U(n). We focus on the particular case \(n=2\), yielding the system of matrix ODE’s on SU(2) with the group manifold \(S^3\). This choice implies restriction on the dimension of multivariate data: in our simulations we investigate data sets where data are represented as vectors in \({\mathbb {R}}^k\), with \(k \le 6\). In our approach each object corresponds to one Kuramoto–Lohe oscillator on \(S^3\) and the data are encoded into matrices of their intrinsic frequencies. We assume global (all-to-all) coupling, which allows to greatly reduce computational cost. One important advantage of this approach is that it can be naturally adapted to clustering of multivariate functional data. We present the simulation results for several illustrative data sets.

Keywords

Data clustering Quantum synchronization Non-Abelian Kuramoto model Kuramoto–Lohe oscillators 

References

  1. Arenas A, Díaz-Guilera A, Pérez-Vicente CJ (2006) Synchronization reveals topological scales in complex networks. Phys Rev Lett 96(11):114102CrossRefGoogle Scholar
  2. Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex networks. Phys Rep 469(3):93–153MathSciNetCrossRefGoogle Scholar
  3. Central Intelligence Agency (2017) The World Factbook. https://www.cia.gov/library/publications/the-world-factbook
  4. Jaćimović V, Crnkić A (2017) Characterizing complex networks through statistics of Möbius transformations. Physica D Nonlinear Phenom 345:56–61CrossRefGoogle Scholar
  5. Jacques J, Preda C (2012) Clustering multivariate functional data. In: COMPSTAT 2012, Cyprus, pp 353–366Google Scholar
  6. Jacques J, Preda C (2014) Model-based clustering for multivariate functional data. Comput Stat Data Anal 71:92–106MathSciNetCrossRefGoogle Scholar
  7. Jain AK (2010) Data clustering: 50 years beyond K-means. Pattern Recognit Lett 31(8):651–666CrossRefGoogle Scholar
  8. Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Comput Surv (CSUR) 31(3):264–323CrossRefGoogle Scholar
  9. Kuramoto Y (1975) Self-entrainment of a population of coupled nonlinear oscillators. In: Proceedings of international symposium on mathematical problems in theoretical physics, pp 420–422Google Scholar
  10. Lohe MA (2009) Non-Abelian Kuramoto models and synchronization. J Phys A Math Theor 42(39):395101MathSciNetCrossRefGoogle Scholar
  11. Lohe MA (2010) Quantum synchronization over quantum networks. J Phys A Math Theor 43(46):465301MathSciNetCrossRefGoogle Scholar
  12. Miyano T, Tsutsui T (2007) Data synchronization in a network of coupled phase oscillators. Phys Rev Lett 98(2):024102CrossRefGoogle Scholar
  13. Novikov AV, Benderskaya EN (2014) Oscillatory neural networks based on the Kuramoto model for cluster analysis. Pattern Recognit Image Anal 24(3):365–371CrossRefGoogle Scholar
  14. Ramsay JO, Silverman BW (2005) Functional data analysis. Springer Series in Statistics. Springer, New YorkCrossRefGoogle Scholar
  15. Rodrigues FA, Peron TKDM, Ji P, Kurths J (2016) The Kuramoto model in complex networks. Phys Rep 610:1–98MathSciNetCrossRefGoogle Scholar
  16. Shao J, He X, Böhm C, Yang Q, Plant C (2013) Synchronization-inspired partitioning and hierarchical clustering. IEEE Trans Knowl Data Eng 25(4):893–905CrossRefGoogle Scholar
  17. Wikipedia contributors (2017,) Iris flower data set—Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Iris_flower_data_set&oldid=815019107
  18. Yamamoto M (2012) Clustering of functional data in a low-dimensional subspace. Adv Data Anal Classif 6(3):219–247MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Faculty of Technical EngineeringUniversity of BihaćBihaćBosnia and Herzegovina
  2. 2.Faculty of Natural Sciences and MathematicsUniversity of MontenegroPodgoricaMontenegro

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