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Quasiorthogonal Dimension

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Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications

Part of the book series: Studies in Computational Intelligence ((SCI,volume 835))

Abstract

An interval approach to the concept of dimension is presented. The concept of quasiorthogonal dimension is obtained by relaxing exact orthogonality so that angular distances between unit vectors are constrained to a fixed closed symmetric interval about \(\pi /2\). An exponential number of such quasiorthogonal vectors exist as the Euclidean dimension increases. Lower bounds on quasiorthogonal dimension are proven using geometry of high-dimensional spaces and a separate argument is given utilizing graph theory. Related notions are reviewed.

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Acknowledgements

V. Kůrková was partially supported by the Czech Grant Foundation grant GA19-05704S and institutional support of the Institute of Computer Science RVO 67985807. P. C. Kainen received research support from Georgetown University.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20057, USA

    Paul C. Kainen

  2. Institute of Computer Science of the Czech Academy of Sciences, Pod Vodárenskou věží  2, 18207, Prague, Czech Republic

    Věra Kůrková

Authors
  1. Paul C. Kainen
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Correspondence to Paul C. Kainen .

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Editors and Affiliations

  1. Department of Teacher Education, University of Texas at El Paso, El Paso, TX, USA

    Olga Kosheleva

  2. Institute of Computational Technologies SB RAS, Novosibirsk, Russia

    Sergey P. Shary

  3. Johns Creek, GA, USA

    Gang Xiang

  4. Deptartment of Informatics, The State Russian Museum, Saint Petersburg, Russia

    Roman Zapatrin

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Kainen, P.C., Kůrková, V. (2020). Quasiorthogonal Dimension. In: Kosheleva, O., Shary, S., Xiang, G., Zapatrin, R. (eds) Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications. Studies in Computational Intelligence, vol 835. Springer, Cham. https://doi.org/10.1007/978-3-030-31041-7_35

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