Quantitative bounds oil rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f E C-d(R-...
详细信息
Quantitative bounds oil rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f E C-d(R-d), with lower-order partials vanishing at infinity and dth-order partials vanishing as parallel to x parallel to(-(d+l+epsilon)), epsilon > 0, on any domain Omega subset of R-d with unit Lebesgue measure, tile L-2(Omega)-error in approximating f by a linear combination of n Heaviside plane waves is bounded above by k(d) parallel to f parallel to d,l,infinity n(-1/2), where k(d) similar to (pi d)(1/2) (e/2 pi)(d/2) and parallel to f parallel to *** is the Sobolev seminorm determined by the largest of the L-l-norms of the dth-order partials off on R-d. In particular, ford odd and f(x) =exp(-parallel to x parallel to(2)), the L-2 (Omega)-approxiination error is at most (2 pi d)(3/4),n(-1/2) and the sup-norm approximation error on R-d is at most 68 root 2(n-1)(-1/2)(2 pi d)(3/4) root d+1, n >= 2 (C) 2007 Elsevier Inc. All rights reserved.
We estimate variation with respect to half-spaces in terms of ''flows through hyperplanes''. Our estimate is derived from an integral representation for smooth compactly supported multivariable functio...
详细信息
We estimate variation with respect to half-spaces in terms of ''flows through hyperplanes''. Our estimate is derived from an integral representation for smooth compactly supported multivariable functions proved using properties of the Heaviside and delta distributions. Consequently we obtain conditions which guarantee approximation error rate of order O(1/root n) by one-hidden-layer networks with n sigmoidal perceptrons. (C) 1997 Elsevier Science Ltd.
We give upper bounds on rates of approximation of real-valued functions of d Boolean variables by one-hidden-layer perceptron networks. Our bounds are of the form c/root n where c depends on certain norms of the funct...
详细信息
We give upper bounds on rates of approximation of real-valued functions of d Boolean variables by one-hidden-layer perceptron networks. Our bounds are of the form c/root n where c depends on certain norms of the function being approximated and n is the number of hidden units. We describe sets of functions where these norms grow either polynomially or exponentially with d. (C) 1998 Elsevier Science Ltd. All rights reserved.
暂无评论