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

Abstract. The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class $ \cal H$ of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width $ \rho_n({\cal F}, L_q) $ = $ inf_{{\cal H}^n} \mbox{dist}({\cal F}, {\cal H}^n, L_q) $ which measures the worst-case approximation error over all functions $ f\in {\cal F} $ by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρn (Wr,dp, Lq) , both being a constant factor of n-r/d , for a Sobolev class Wr,dp , $ 1 \leq p, q \leq \infty $ . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of Wr,dp 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.