Relations between Classical, Average, and Probabilistic Kolmogorov Widths

Let ? be a centered Gaussian Radon measure on a Banach space E, and let H??E be its reproducing kernel Hilbert space with unit ball K?. We prove that for the ?-average widths d(a)n(E, ?) of E and the classical Kolmogorov widths dn(K?, E) we haved(a)n(E, ?)?n??(logn)siffdn(K?, E)?n?1/2??(logn)s for any ?>0, s?R. Moreover, order optimal subspaces for dn(K?, E) are order optimal for d(a)n(E, ?) as well. Furthermore, we show that for the probabilistic widths d(p)n, ?(E, ?) we have the estimate12d(a)n(E, ?)?d(p)n, ?(E, ?)?c 1d(a)n(E, ?)(1+log2/?) for some universal constant c1>0 and for all ?<?0. These results are applied to find concrete estimates in some specific settings.

[1]  Vitaly Maiorov.  About Widths of Wiener Space in the Lq-Norm , 1996, J. Complex..

[2]  Dualité des nombres d'entropie pour des opérateurs à valeurs dans un espace de Hilbert , 1987 .

[3]  Werner Linde,et al.  Mappings of Gaussian Cylindrical Measures in Banach Spaces , 1975 .

[4]  J. Kuelbs,et al.  Metric entropy and the small ball problem for Gaussian measures , 1993 .

[5]  Klaus Ritter,et al.  Average-case analysis of numerical problems , 2000, Lecture notes in mathematics.

[6]  Entropy of C(K)-valued operators and some applications , 2000 .

[7]  R. Dudley.  The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes , 1967 .

[8]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[9]  Thomas Kühn,et al.  Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet , 1999 .

[10]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[11]  KOLMOGOROV'S $ (n,\,\delta)$-WIDTHS OF SPACES OF SMOOTH FUNCTIONS , 1994 .

[12]  Q. Shao,et al.  Gaussian processes: Inequalities, small ball probabilities and applications , 2001 .

[13]  Werner Linde,et al.  Approximation, metric entropy and small ball estimates for Gaussian measures , 1999 .

[14]  H. Triebel,et al.  Function Spaces, Entropy Numbers, Differential Operators: Function Spaces , 1996 .

[15]  Vitaly Maiorov,et al.  Average n-Widths of the Wiener Space in the Linfinity-Norm , 1993, J. Complex..

[16]  B. Carl,et al.  Entropy, Compactness and the Approximation of Operators , 1990 .

[17]  Small Deviations for Gaussian Markov Processes Under the Sup-Norm , 1999 .

[18]  B. Carl,et al.  s-Numbers of integral operators with Hölder continuous kernels over metric compacta , 1988 .