Geodesic
Widths related to pseudo-dimension
Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 319-332.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider two widths related to the notion of pseudo-dimension. The first is $\rho_n$, which is defined in a similar way to Kolmogorov's width but replacing the linear dimension by the pseudo-dimension. $\rho_n$ can be bounded below by the second width $s_n$, which is half of the length of the maximal edge of the $(n+1)$-dimensional ‘coordinate’ cube inscribed in the given set in a special way. We construct examples of sets for which the ratios $\rho_n/s_n$ (for $n\geqslant 2$) and $\rho_{10n}/s_{9n}$ (for a sufficiently large $n$) are as large as desired. In terms of combinatorial dimension, the main result means that for any $C>0$ and any sufficiently large $n$ there is a set $W$ of dimension $\mathrm{vc}(W,1)\leqslant 9n$ which cannot be approximated with respect to the uniform norm with accuracy $C$ by any set $V$ of dimension $\mathrm{vc}(V,0)\leqslant 10n$.
Keywords: combinatorial dimension, widths.
Mots-clés : VC-dimension 
@article{IM2_2009_73_2_a3,
     author = {Yu. V. Malykhin},
     title = {Widths related to pseudo-dimension},
     journal = {Izvestiya. Mathematics },
     pages = {319--332},
     publisher = {mathdoc},
     volume = {73},
     number = {2},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/}
}
TY  - JOUR
AU  - Yu. V. Malykhin
TI  - Widths related to pseudo-dimension
JO  - Izvestiya. Mathematics 
PY  - 2009
SP  - 319
EP  - 332
VL  - 73
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/
LA  - en
ID  - IM2_2009_73_2_a3
ER  - 
%0 Journal Article
%A Yu. V. Malykhin
%T Widths related to pseudo-dimension
%J Izvestiya. Mathematics 
%D 2009
%P 319-332
%V 73
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/
%G en
%F IM2_2009_73_2_a3
Yu. V. Malykhin. Widths related to pseudo-dimension. Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 319-332. http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/

[1] V. N. Vapnik, A. Ya. Chervonenkis, “On the uniform convergence of relative frequencies of events to their probabilities”, Theor. Probability Appl., 16:2 (1971), 264–280 | DOI | MR | Zbl

[2] H. S. Shapiro, “Some negative theorems of approximation theory”, Michigan Math. J., 11:3 (1964), 211–217 | DOI | MR | Zbl

[3] H. E. Warren, “Lower bounds for approximation by nonlinear manifolds”, Trans. Amer. Math. Soc., 133:1 (1968), 167–178 | DOI | MR | Zbl

[4] A. Andrianov, “On pseudo-dimension of certain sets of functions”, East J. Approx., 5:4 (1999), 393–402 | MR | Zbl

[5] V. Maiorov, J. Ratsaby, “On the degree of approximation by manifolds of finite pseudo-dimension”, Constr. Approx., 15:2 (1999), 291–300 | DOI | MR | Zbl

[6] V. Maiorov, “Optimal non-linear approximation using sets of finite pseudo-dimension”, East J. Approx., 11:1 (2005), 1–19 | MR | Zbl

[7] S. Mendelson, R. Vershynin, “Entropy and the combinatorial dimension”, Invent. Math., 152:1 (2003), 37–55 | DOI | MR | Zbl

[8] M. Rudelson, R. Vershynin, “Combinatorics of random processes and sections of convex bodies”, Ann. of Math. (2), 164:2 (2006), 603–648 | DOI | MR | Zbl