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Generalised rational approximation and its application to improve deep learning classifiers

Author

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  • Peiris, V.
  • Sharon, N.
  • Sukhorukova, N.
  • Ugon, J.

Abstract

A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals.

Suggested Citation

  • Peiris, V. & Sharon, N. & Sukhorukova, N. & Ugon, J., 2021. "Generalised rational approximation and its application to improve deep learning classifiers," Applied Mathematics and Computation, Elsevier, vol. 389(C).
    • Handle: RePEc:eee:apmaco:v:389:y:2021:i:c:s0096300320305166
    DOI: 10.1016/j.amc.2020.125560
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    References listed on IDEAS

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    1. Zamir, Z. Roshan & Sukhorukova, N. & Amiel, H. & Ugon, A. & Philippe, C., 2015. "Convex optimisation-based methods for K-complex detection," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 947-956.
    2. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    3. Roshan Zamir, Z. & Sukhorukova, N., 2016. "Linear least squares problems involving fixed knots polynomial splines and their singularity study," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 204-215.
    4. Jean-Pierre Crouzeix & Nadezda Sukhorukova & Julien Ugon, 2017. "Characterization Theorem for Best Polynomial Spline Approximation with Free Knots, Variable Degree and Fixed Tails," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 950-964, March.
    5. Nadezda Sukhorukova, 2010. "Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 378-394, November.
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