On the Approximation of Functional Classes Equipped with a Uniform Measure Using Ridge Functions

We introduce a construction of a uniform measure over a functional class B which is similar to a Besov class with smoothness index r. We then consider the problem of approximating B using a manifold Mn which consists of all linear manifolds spanned by n ridge functions, i.e., Mn=[ i=1 gi(ai } x) : ai # S , gi # L2([&1, 1])], x # Bd. It is proved that for some subset A/Br of probabilistic measure 1&$, for all f # A the degree of approximation of Mn behaves asymptotically as 1 n . As a direct consequence the probabilistic (n, $ )-width for nonlinear approximation denoted as dn, $ (B, +, Mn), where + is a uniform measure over B, is similarly bounded. The lower bound holds also for the specific case of approximation using a manifold of one hidden layer neural networks with n hidden units. 1999 Academic Press