Random approximants and neural networks
Let \(D\) be a set with a probability measure \(\mu\), \(\mu(D)=1\), and let \(K\) be a compact subset of \(L_q(D,\mu)\), \(1\leq q<\infty\). For \(f\in L_q\), \(n=1,2,\dots\), let \(\rho_n(f,K)=\inf|f-g_n|_q\), where the infimum is taken over all \(g_n\) of the form \(g_n= \sum^n_{i=1}a_i\phi_i\), with arbitrary \(\phi_i\in K\) and \(a_i\in R\). It is shown that for \(f\in \text{conv}(\pm K)\), under some mild restrictions, \(\rho_n(f,K)\leq C_q\varepsilon_n(K)n^{-1/2}\), where \(\varepsilon_n(K)\to 0\) as \(n\to\infty\). This fact is used to estimate the errors of certain neural net approximations. For the latter, also the lower estimates of errors are given.
scientific article; zbMATH DE number 885473