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^r which is similar to a Besov class with smoothness index r. We then consider the problem of approximating B^r using a manifold M_n which consists of all linear manifolds spanned by n ridge functions, i.e., M_n={∑ⁿ_i=1g_i(a _i·x):a_i∈S^d-1, g_i∈L₂([-1, 1])}, x∈B^d. It is proved that for some subset A⊂B^r of probabilistic measure 1-δ, for all f∈A the degree of approximation of M_n behaves asymptotically as 1/n^r/(d-1). As a direct consequence the probabilistic (n, δ)-width for nonlinear approximation denoted as d_{n, δ}(B^r, μ, M_n), where μ is a uniform measure over B^r, 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.