Burton Robert M., Dehling Herold G.
Department of Mathematics, Oregon State University, Corvallis, USA
Neural Netw. 1998 Jun;11(4):661-667. doi: 10.1016/s0893-6080(98)00009-4.
A feedforward neural net with d input neurons and with a single hidden layer of n neurons is given byg(x(1), em leader,x(d))= summation operator j=1na(j)sigma,where a(j), theta(j), w(ji) in R. In this paper we study the approximation of arbitrary functions F:R(d)-->R by a neural net in an L(p)(&mgr;) norm for some finite measure &mgr; on R(d). We prove that under natural moment conditions, a neural net with non-polynomial function can approximate any given function.
一个具有(d)个输入神经元且只有一层包含(n)个神经元的隐藏层的前馈神经网络由(g(x^{(1)},\cdots,x^{(d)})=\sum_{j = 1}^{n}a^{(j)}\sigma(\sum_{i = 1}^{d}w^{(ji)}x^{(i)}+\theta^{(j)}))给出，其中(a^{(j)},\theta^{(j)},w^{(ji)}\in\mathbb{R})。在本文中，我们研究对于(\mathbb{R}^{d})上的某个有限测度(\mu)，神经网络在(L^{p}(\mu))范数下对任意函数(F:\mathbb{R}^{d}\to\mathbb{R})的逼近。我们证明，在自然矩条件下，具有非多项式函数的神经网络可以逼近任何给定函数。
