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 class H of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width rho(n)(F, L-q) = inf(H)(n) dist(F, H-n. L-q,,) which measures the worstcase approximation error over all functions f is an element of F by the best manifold of pseudo-dimension n. In this paper we obtain tight upper and lower bounds on p(n)(W-p(r.d), L-q), both being a constant factor of n(-r/d), for a Sobolev class W-p(r.d), 1 less than or equal to p.q less than or equal to infinity. As this is also the estimate of the classical Alexandrov nonlinear n-width, our result proves that approximation of W-p(r.d) 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.