TY - JOUR

T1 - The degree of approximation of sets in euclidean space using sets with bounded Vapnik-Chervonenkis dimension

AU - Maiorov, Vitaly

AU - Ratsaby, Joel

N1 - Funding Information:
J. Ratsaby acknowledges the support of a VATAT Post-Doctorate fellowship and the Ollendorff center of the Department of Electrical Engineering at the Technion. V. Maiorov acknowledges the support of the center for absorption in science of the minis~ of i~igr~t abso~tion, state of Israel. He also ac~owledges the support of Allan Pinkus of the Department of Mathematics, Technion. The authors thank the reviewers for interesting and informative comments.

PY - 1998/8/18

Y1 - 1998/8/18

N2 - The 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)- supx∈F infy∈Hn ∥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 ∈ 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 1/2 < γ < 1.

AB - The 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)- supx∈F infy∈Hn ∥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 ∈ 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 1/2 < γ < 1.

UR - http://www.scopus.com/inward/record.url?scp=0002702217&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(98)00015-8

DO - 10.1016/S0166-218X(98)00015-8

M3 - Article

AN - SCOPUS:0002702217

SN - 0166-218X

VL - 86

SP - 81

EP - 93

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1

ER -