The degree of approximation of sets in euclidean space using sets with bounded Vapnik-Chervonenkis dimension
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 ρ n ( F, L q ) which measures the degree of app...
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| Vydané v: | Discrete Applied Mathematics Ročník 86; číslo 1; s. 81 - 93 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Lausanne
Elsevier B.V
18.08.1998
Amsterdam Elsevier New York, NY |
| Predmet: | |
| ISSN: | 0166-218X, 1872-6771 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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
ρ
n
(
F,
L
q
) which measures the degree of approximation of a function class
F by the best manifold
H
n
of pseudo-dimension less than or equal to
n in the
L
q
-metric has been introduced. For sets
F ⊂
R
m
it is defined as
ρ
n
(
F,
l
m
q
) = inf
H
n
dist(
F,
H
n
), where
dist(
F,
H
n
) = sup
xϵF
inf
yϵH
n
∥
x−
y ∥
l
m
q
and
H
n ⊂
R
m
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
H
n ⊂
R
m
of VC-dimension less than or equal to
n in the
l
m
q
-metric. In this paper we compute
ρ
n
(
F,
l
m
q
) for
F being the unit ball
B
m
p = {x ϵ
R
m : ∥x∥
l
m
p
⩽ 1}
for any 1 ⩽
p,
q ⩽ ∞, and for
F being any subset of the boolean
m-cube of size larger than 2
mγ
, for any
1
2
<γ< 1
. |
|---|---|
| ISSN: | 0166-218X 1872-6771 |
| DOI: | 10.1016/S0166-218X(98)00015-8 |