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

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=1 gi(ai·x): ai∈Sd−1, gi∈L2([−1, 1])}, x∈Bd. It is proved that for some subset A⊂Br of probabilistic measure 1−δ, for all f∈A the degree of approximation of Mn behaves asymptotically as 1/nr/(d−1). As a direct consequence the probabilistic (n, δ)-width for nonlinear approximation denoted as dn, δ(Br, μ, Mn), where μ 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.