Gel ' fand n - Widths and the Method of Least Squares

Consider the problem of estimating a function f known to lie in a convex, compact subset F of L2[0, 1], when one observes data on f containing white Gaussian noise. We establish an upper bound on the integrated mean-squared error of least-squares estimates which uses the asymptotic properties of the Gel'fand n-widths of F. The bound shows that if the Gel'fand n-widths tend to zero at a faster rate than the Kolmogorov linear n-widths, least squares must outperform every orthogonal series estimator, at the level of minimax rates of convergence. NVe rely heavily on Carl's Theorem, a recent development in the Geometry of Banach Spaces. As an application, we resolve a question about the performance of least-squares estimates in estimating decreasing functions from noisy sampled data. Acknowledgements. The author would like to thank Lucien Birge for helpful conversations at the Ecole d'Ete de Probabilites in Saint Flour, and to Prof. P.L. Hennequin for arranging my visit there. Conversations with Henryk Wozniakowski about Gel'fand n-widths were also helpful. This work was supported by NSF DMS-88-10192, and by NASA NCA2-0488.