Uniform approximation by neural networks
Let \(X\) be a Banach space, let \(\Phi \subset X\) be a bounded set, and let \(f=\sum_k c_k\phi_k\), \(\phi_k\in\Phi\), \(c_k\in {\mathbb R}\), where the sum can be finite or infinite or more generally, \(f\) can be of the form \(f=\int c_{\lambda}\phi_{\lambda}d\mu(\lambda)\), in an appropriate setting. Given a natural number \(n\), it is required to find a good approximation to \(f\) by a linear combination \(g_n =\sum_{i=1}^n a_i\phi_{k_i}\) of at most \(n\) of the \(\phi_k\). The author previously considered [J. Approximation Theory 41, 361-366 (1984; Zbl 0543.41027)] this problem for \(X=L_q(D)\), \(q<\infty\), where \(D\) is a compact set in \({\mathbb R^n}\), assuming that the set \(\Phi\) is bounded in \(L_{\infty}(D)\). In this paper, he obtains by probabilistic methods an estimate for \(\|f - g_n\|\) in the uniform norm (i.e. \(X=L_{\infty}(D)\)). Two main cases are considered: (1) where \(D\) is a finite set and (2) where the set \(\Phi\) is formed by the functions \(\phi_{v,b}(x)\colon=s(v\cdot x+b)\), where \(v\cdot x\) is the scalar product, \(v,x\in {\mathbb R^d}\), \(b\in {\mathbb R}\), and \(s\) is a fixed \({\mathbb R} \to{\mathbb R}\) function.
scientific article; zbMATH DE number 1230422