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 irtroduced a nonlinear approximation width ρ_n(F, L_q) = inf_Hn dist(F, Hⁿ, L_q) which measures the worst-case approximation error over all functions f ∈ F by the best manifold of pseudo-dimension n. In this paper we obtain tight upper and lower bounds on ρ_n(W^r,d_p, L_q), both being a constant factor of n^-r/d, for a Sobolev class W^r,d_p. l ≤ 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.