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The Intrinsic Dimensionality of Data

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Abstract

We consider the problem of determining the intrinsic dimensionality of data which is important for optimizing the organization and processing of large data sets in classical machines, quantum decision theory, and observations of natural phenomena. We prove a theorem that determines the minimum dimensions associated with the data and this result is consistent with the result that base-e is optimal for number representation. The dimension value be viewed as coding the structure in the most efficient representation of the data and has relevance for natural and engineered systems. Since the optimal intrinsic dimensionality is shown to be noninteger, this paper provides a rationale for fractals in natural data.

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References

  1. D. Aerts, Quantum structure in cognition. J. Math. Psychol. 53, 314–348 (2009)

    Article  MathSciNet  Google Scholar 

  2. J.R. Armstrong et al., Analytic solutions of topologically disjoint systems. J. Phys. A: Math. Theor. 48, 085301 (2015)

    Article  MathSciNet  Google Scholar 

  3. M. Baker, 1500 scientists lift the lid on reproducibility. Nature 533, 452–454 (2016)

    Article  Google Scholar 

  4. M. Baker, E. Dolgin, Cancer reproducibility project releases first results. Nature 541, 269–270 (2017)

    Article  Google Scholar 

  5. A. Bunde, S. Havlin, Fractals in Science (Springer, Berlin, 2013)

    MATH  Google Scholar 

  6. J.R. Busemeyer, P. Bruza, Quantum Models of Cognition and Decision (Cambridge University Press, Cambridge, 2012)

    Book  Google Scholar 

  7. A. Carpinteri, F. Mainardi (eds.), Fractals and Fractional Calculus in Continuum Mechanics (Springer, Wien, 1997)

    MATH  Google Scholar 

  8. R.C. Conant, W.R. Ashby, Every good regulator of a system must be a model of that system. Int. J. Syst. Sci. 1(2), 89–97 (1970)

    Article  MathSciNet  Google Scholar 

  9. K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Hoboken, 2003)

    Book  Google Scholar 

  10. S. Kak, On training feedforward neural networks. Pramana 40, 35–42 (1993)

    Article  Google Scholar 

  11. S. Kak, Faster web search and prediction using instantaneously trained neural networks. IEEE Intell. Syst. 14, 79–82 (1999)

    Google Scholar 

  12. S. Kak, Communication languages and agents in biological systems, in Biocommunication: Sign-Mediated Interactions between Cells and Organisms, ed. by R. Gordon, J. Seckbach (World Scientific Publishing, London, 2016), pp. 203–226

    Google Scholar 

  13. S. Kak, State ensembles and quantum entropy. Int. J. Theor. Phys. 55, 3017–3026 (2016)

    Article  Google Scholar 

  14. S. Kak, Incomplete information and quantum decision trees. in IEEE SMC 2017, International Conference on Systems, Man, and Cybernetics. Banff, Canada (2017)

  15. S. Kak, Learning Based on CC1 and CC4 Neural Networks. arXiv:1712.09331 (2017)

  16. S. Kak, The base-e representation of numbers and the power law. Circuits Syst. Signal Process. (2020). https://doi.org/10.1007/s00034-020-01480-0

    Article  Google Scholar 

  17. S. Kak, Information, representation, and structure. in International Conference on Recent Trends in Mathematics and Its Applications to Graphs, Networks and Petri Nets, New Delhi, India (2020)

  18. S. Kak, Noninteger dimensional spaces and the inverse square law. (2020). TechRxiv: https://www.techrxiv.org/articles/preprint/Noninteger_Dimensional_Spaces_and_the_Inverse_Square_Law/13079720

  19. A.Y. Khrennikov, Ubiquitous Quantum Structure: From Psychology to Finance (Springer, Berlin, 2010)

    Book  Google Scholar 

  20. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  21. A. Kwiatkowski, H. Werner, PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding. IEEE Trans. Control Syst. Technol. 16, 781–788 (2008)

    Article  Google Scholar 

  22. B.B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983)

    Book  Google Scholar 

  23. E.R. Omiecinski, Alternative interest measures for mining associations in databases. IEEE Trans. Knowl. Data Eng. 15, 57–69 (2003)

    Article  Google Scholar 

  24. R. Panek, A cosmic crisis. Sci. Am. 322(3), 30–37 (2020)

    Google Scholar 

  25. A. Shortt, J.G. Keating, L. Moulinier, C.N. Pannell, Optical implementation of the Kak neural network. Inf. Sci. 171, 273–287 (2005)

    Article  MathSciNet  Google Scholar 

  26. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimensions. J. Math. Phys. 18, 1224–1234 (1977)

    Article  MathSciNet  Google Scholar 

  27. K.W. Tang, S. Kak, Fast classification networks for signal processing. Circuits, Syst. Signal Process. 21, 207–224 (2002)

    Article  Google Scholar 

  28. V.E. Tarasov, Anisotropic fractal media by vector calculus in non-integer dimensional space. J. Math. Phys. 55, 083510 (2014)

    Article  MathSciNet  Google Scholar 

  29. W.P. Thurston, Orbifolds, in The Geometry and Topology of Three-Manifolds (Princeton University Press, Princeton, 1997), pp. 297–355

    Google Scholar 

  30. M. Verleysen, E. de Bodt, A. Lendasse, Forecasting financial time series through intrinsic dimension estimation and non-linear data projection. in Proceedings of IWANN’99—International Work-conference on Artificial and Natural Neural Networks, Alicante (Spain), June 2–4, 1999, Springer, Lecture Notes in Computer Science 1607, J. Mira, Juan V. Sanchez-Andres eds. (1999)

  31. G.I. Webb, Discovering significant rules. in Proceedings of the Twelfth ACM SIGKDD International Conference on Knowledge Discovery and Data mining, KDD-2006. (pp. 434–443). New York, NY: ACM (2006)

  32. S. Wolfram, A Class of Models with the Potential to Represent Fundamental Physics. arXiv:2004.08210 (2020)

  33. V.I. Yukalov, D. Sornette, Decision theory with prospect interference and entanglement. Theor. Decis. 70, 283–328 (2010)

    Article  MathSciNet  Google Scholar 

  34. L. Zhou, Chromatic numbers of the Menger sponges. Am. Math. Mon. 114(9), 842 (2007)

    Google Scholar 

  35. J. Zhu, G. Milne, Implementing kak neural networks on a reconfigurable computing platform. Lect. Notes Comput. Sci. 1896, 260–269 (2000)

    Article  Google Scholar 

Download references

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  1. Oklahoma State University, Stillwater, USA

    Subhash Kak

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  1. Subhash Kak
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Correspondence to Subhash Kak.

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Kak, S. The Intrinsic Dimensionality of Data. Circuits Syst Signal Process 40, 2599–2607 (2021). https://doi.org/10.1007/s00034-020-01583-8

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  • DOI: https://doi.org/10.1007/s00034-020-01583-8

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