Lower bounds for multivariate approximation by affine-invariant dictionaries

The problem of approximating locally smooth multivariate functions by linear combinations of elements from an affine-invariant redundant dictionary is considered. Augmenting previous upper bound results for approximation, we establish lower bounds on the performance of such schemes. The lower bounds are tight to within a logarithmic factor in the number of elements used in the approximation. Using a previously introduced notion of nonlinear approximation, we show that the approximation ability may be completely characterized by the pseudodimension of the approximation space with respect to a finite set of points. This result establishes a useful link between the problems of approximation and estimation, or learning, the latter often being conveniently characterized, at least in terms of upper bounds, by the pseudodimension.