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 Br which is similar to a Besov class with smoothness index r. We then consider the problem of approximating Br using a manifold Mn which consists of all linear manifolds spanned by n ridge functions, i.e., Mn={?ni=1gi(ai·x):ai?Sd?1, 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/nr/(d?1). As a direct consequence the probabilistic (n, ?)-width for nonlinear approximation denoted as dn, ?(Br, ?, Mn), where ? is a uniform measure over Br, 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.