An approximation by neural networkswith a fixed weight

On the approximation by single hidden layer feedforward neural networks with fixed weights

“…We investigate the sigmoidal neural network approximation to f (x). This function was also considered in [13]. Note that in [13] the authors chose the sigmoidal function as…”

“…In many applications, it is convenient to take the activation function σ as a sigmoidal function which is defined as lim t→−∞ σ(t) = 0 and lim t→+∞ σ(t) = 1. The literature on neural networks abounds with the use of such functions and their superpositions (see, e.g., [2,4,6,8,10,11,13,15,20,22,29]). The possibility of approximating a continuous function on a compact subset of the real line or d-dimensional space by SLFNs with a sigmoidal activation function has been well studied in a number of papers.…”

“…For a set W of weights consisting of two directions, he showed that there is a geometrically explicit solution to the problem. Hahm and Hong [13] went further in this direction, and showed that SLFNs with fixed weights can approximate arbitrarily well any univariate function. Since fixed weights reduce the computational expense and training time, this result is of particular interest.…”

“…In a mathematical formulation, the result reads as follows. Theorem 1.1 (Hahm and Hong [13]). Assume f is a continuous function on a finite segment [a, b] of R. Assume σ is a bounded measurable sigmoidal function on R. Then for any sufficiently small ε > 0 there exist constants c i , θ i ∈ R and positive integers K and n such that…”

The Capability of Localized Neural Network Approximation

“…Many mathematicians ( [2], [4], [5], [6], [7], [9], [10], [11]) have been studied the neural network approximation in recent years. In [1], Chui, Li and Mhaskar pointed out the limitation of approximation by neural network with one hidden layer.…”