基于有理逼近与神经网络的逼近方法之比较

Rational approximation and its application to improving deep learning classifiers

A rational approximation by a ratio of polynomial functions 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 are quasiconvex, and show how to use this fact for calculating the best approximation in a fast and efficient method. The paper presents a theoretical study of the arising optimisation problems and provides results of several numerical…

The extension of linear inequality method for generalised rational Chebyshev approximation to approximation by general quasilinear functions

In this paper we demonstrate that a well known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimisation. Although this correspondence is not surprising, it naturally connects rational and generalised rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and therefore offers more theoretical and computational tools for solving this problem. %In particular, this observation leads a straightforward extension of the linear inequality method to g…

Rational approximations of spectral densities based on the Alpha divergence

We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by minimizing a suitable distance from the given spectrum and under the constraints corresponding to imposing the given second-order statistics. Here, we consider the Alpha divergence family as a distance measure. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback-Leibler solution proposed by Georgiou and Lindquist in 2003. Fina…

多元多项式与广义有理函数逼近

Multivariate approximation by polynomial and generalised rational functions

本文提出基于优化的多元 Chebyshev 逼近方法，证明广义有理逼近的优化问题为拟凸，可用二分法与线性规划求解，并比较了多项式与广义有理逼近的误差与计算时间。

An algorithm for best generalised rational approximation of continuous functions

The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous function. It is known that the objective functions in generalised rational approximation problems are quasi-convex. In this paper we also prove a stronger result, the objective functions are pseudo-…

Rational Polyhedral Outer-Approximations of the Second-Order Cone

It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy $ε$ by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral outer-approximations of compact size retaining the same guaranteed accuracy $ε$. The first outer-approximation has the same size as the optimal but irrational outer-approximation from the literature. In this case,we provide a practical approach to obtain such an approximation defined by the smallest integer coefficients possible, which requires solving a few, small-size integer quadr…