Relations between classical, average, and probabilistic Kolmogorov widths

Abstract

Let μ be a centered Gaussian Radon measure on a Banach space E, and let Hμ ⊆ E be its reproducing kernel Hilbert space with unit ball Kμ. We prove that for the μ-average widths dn(a)(E, μ) of E and the classical Kolmogorov widths dn(Kμ, E) we have for any α > 0, β ε R. Moreover, order optimal subspaces for dn(Kμ, E) are order optimal for dn(a)(E, μ) as well. Furthermore, we show that for the probabilistic widths dn,δ(p)(E, μ) we have the estimate for some universal constant c1 > 0 and for all δ < δ0. These results are applied to find concrete estimates in some specific settings. © 2002 Elsevier Science (USA).