The author studies geometrical properties of the ridge function manifold \(\mathcal{R}_n\) consisting of all possible linear combinations of \(n\) functions of the form \(g(a\cdot x)\), where \(a\cdot x\) is the inner product in \({\mathbb R}^d\)and obtains an estimate for the \(\varepsilon \)-entropy numbers in terms of smaller \(\varepsilon \)-covering numbers of the compact class \(G _{n,s }\) formed by the intersection of the class \(\mathcal{R}_n\) with the unit ball \(B\mathcal{P}_s^d\) in the space of polynomials on \({\mathbb R}^d\) of degree \(s\). In particular he shows that for \(n \leq s ^{d - 1}\) the \(\varepsilon \)-entropy number \(H _{\varepsilon }(G _{n,s },L _{q })\) of the class \(G _{n,s }\) in the space \(L _{q }\) is of order \(ns \log 1/\varepsilon \) (modulo a logarithmic factor). Note that the \(\varepsilon \)-entropy number \(H_\varepsilon(B\mathcal{P}_s^d,L_q)\) of the unit ball is of order \(s ^{d }\log 1/\varepsilon \). This result answers a question posed by Allan Pinkus: what is the cardinality of the intersection of the ridge function manifold with the unit ball in the space of polynomials of aa given degree?
The author also obtains an estimate for the pseudo-dimension of the ridge function class \(G _{n,s }\).