Relations between classical, average, and probabilistic Kolmogorov widths (Q1599204)

Let \(\mu \) be a centered Gaussian Radon measure on a Banach space \(E\), and let \(H_\mu \subseteq E\) be its reproducing kernel Hilbert space with unit ball \(K_\mu \). The authors prove that for \(\mu\)-average widths \(d_n^{(\alpha)} (E,\mu)\) of \(E\) and the classical Kolmogorov widths \(d_n (K_\mu,E)\) the order relation \(d_n^{(\alpha)} (E,\mu) \approx n^{ -\alpha}(\log n)^\beta\) is valid, iff \(d_n (K_\mu,E) \approx n^{-1/2-\alpha}(\log n)^\beta \) for any \(\alpha > 0\), \(\beta \in R\). Moreover, order optimal subspaces for \(d_n (K_\mu,E)\) are order optimal for \(d_n^{(\alpha)} (E,\mu)\) as well. Furthermore, it is shown that for the probabilistic widths \(d_{n, \delta}^{(p)} (E,\mu)\) the estimation \[ \tfrac 12 d_n^{(\alpha)} (E,\mu)\leq d_{n, \delta }^{(p)} (E,\mu)\leq d_n^{(\alpha)} (E,\mu) ( 1 + \sqrt {\log 2/\delta}) \] for some universal constant \(c_1 > 0\) and for all \(\delta < \delta_0 \) holds. These results are applied to find concrete estimates in some specific settings.