The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural ne...The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.展开更多
Using some regular matrices we present a method to express any multivariate algebraic polynomial of total order n in a normal form. Consequently, we prove constructively that, to approximate continuous target function...Using some regular matrices we present a method to express any multivariate algebraic polynomial of total order n in a normal form. Consequently, we prove constructively that, to approximate continuous target functions defined on some compact set of Rd, neural networks are at least as good as algebraic polynomials.展开更多
In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation fr...In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation from the radial function manifolds to WP^r（b^d）. Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.展开更多
In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation ...In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial An （g） converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.展开更多
基金Supported by National Natural Science Foundation of China （Grant No. 60873206） and Natural Science Foun- dation of Zhejiang Province of China （Grant No. Y7080235）
文摘The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.
文摘Using some regular matrices we present a method to express any multivariate algebraic polynomial of total order n in a normal form. Consequently, we prove constructively that, to approximate continuous target functions defined on some compact set of Rd, neural networks are at least as good as algebraic polynomials.
基金supported by the National 973 Project (Grant No. 2007CB311002)National Natural Science Foundation of China (Grant Nos. 90818020,60873206)
文摘In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation from the radial function manifolds to WP^r（b^d）. Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.
基金Supported by National Natural Science Foundation of China （Grant No. 61063020）, Ningxia Ziran （Grant No. NZ0907） and 2009 Ningxia Gaoxiao Foundations We would like to thank two anonymous referees for their insightful comments.
文摘In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial An （g） converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.
