Mhaskar H N
Department of Mathematics, California State University, Los Angeles, CA 90032, USA.
Neural Netw. 2004 Sep;17(7):989-1001. doi: 10.1016/j.neunet.2004.04.001.
Let s > or = 1 be an integer. A Gaussian network is a function on Rs of the form [Formula: see text]. The minimal separation among the centers, defined by (1/2) min(1 < or = j not = k < or = N) [Formula: see text], is an important characteristic of the network that determines the stability of interpolation by Gaussian networks, the degree of approximation by such networks, etc. Let (within this abstract only) the set of all Gaussian networks with minimal separation exceeding 1/m be denoted by Gm. We prove that for functions [Formula: see text] such that [Formula: see text], if the degree of L2(nonlinear) approximation of [Formula: see text] from Gm is [Formula: see text] then necessarily the degree of approximation of [Formula: see text] by (rectangular) partial sums of degree m2 of the Hermite expansion of [Formula: see text] is also [Formula: see text]. Moreover, Gaussian networks in Gm having fixed centers in a ball of radius [Formula: see text] and coefficients being linear functionals of [Formula: see text] can be constructed to yield the same degree of approximation. Similar results are proved for the Lp norms, 1 < or = p < or =[Formula: see text] but with the condition that the number of neurons N, should satisfy logN = Formula: see text.
设(s\geq1)为整数。高斯网络是(\mathbb{R}^s)上形如[公式：见原文]的函数。由((1/2)\min_{1\leq j\neq k\leq N}[公式：见原文])定义的中心之间的最小间距是该网络的一个重要特征，它决定了高斯网络插值的稳定性、此类网络的逼近程度等。（仅在本摘要范围内）将最小间距超过(1/m)的所有高斯网络的集合记为(G_m)。我们证明，对于满足[公式：见原文]的函数[公式：见原文]，如果从(G_m)对[公式：见原文]的(L^2)（非线性）逼近度为[公式：见原文]，那么必然地，由[公式：见原文]的埃尔米特展开的(m^2)次（矩形）部分和对[公式：见原文]的逼近度也为[公式：见原文]。此外，可以构造(G_m)中的高斯网络，其中心固定在半径为[公式：见原文]的球内且系数是[公式：见原文]的线性泛函，以产生相同的逼近度。对于(1\leq p\leq[公式：见原文])的(L^p)范数也证明了类似结果，但条件是神经元数量(N)应满足(\log N = 公式：见原文)。
