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.

中文翻译：

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广义有理逼近及其在改进深度学习分类器中的应用

摘要 有理逼近（即通过两个多项式的比值逼近）是多项式逼近的灵活替代方案。特别是，有理函数对非光滑和非 Lipschitz 函数表现出准确的估计，其中多项式逼近效率不高。我们证明，即使在基函数不限于单项式时，出现在最佳均匀有理近似及其对基函数线性组合比率的泛化中的优化问题也是拟凸的。然后我们将展示如何在计算方法的开发中使用这一事实。本文对出现的优化问题进行了理论研究，并提供了几个数值实验的结果。