The degree of approximation of sets in euclidean space using sets with bounded Vapnik-Chervonenkis dimension

AbstractThe degree of approximation of infinite-dimensional function classes using finite n-dimensional manifolds has been the subject of a classical field of study in the area of mathematical approximation theory. In Ratsaby and Maiorov (1997), a new quantity ρn(F, Lq) which measures the degree of approximation of a function class F by the best manifold Hn of pseudo-dimension less than or equal to n in the Lq-metric has been introduced. For sets F ⊂Rm it is defined as ρn(F, lmq) = infHn dist(F, Hn), where dist(F, Hn) = supxϵF infyϵHn∥x−y ∥lmq and Hn ⊂Rm is any set of VC-dimension less than or equal to n where n<m. It measures the degree of approximation of the set F by the optimal set Hn ⊂Rm of VC-dimension less than or equal to n in the lmq-metric. In this paper we compute ρn(F, lmq) for F being the unit ball Bmp = {x ϵ Rm : ∥x∥lmp⩽ 1} for any 1 ⩽ p, q ⩽ ∞, and for F being any subset of the boolean m-cube of size larger than 2mγ, for any 12 <γ< 1