On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A classH of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width ρn(F , Lq ) = infHn dist(F ,Hn, Lq ) which measures the worstcase approximation error over all functions f ∈ F by the best manifold of pseudodimension n. In this paper we obtain tight upper and lower bounds on ρn(W r,d p , Lq ), both being a constant factor of n−r/d , for a Sobolev class W r,d p , 1 ≤ p, q ≤ ∞. As this is also the estimate of the classical Alexandrov nonlinear n-width, our result proves that approximation of W r,d p by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators.