Estimates of covering numbers of convex sets with slowly decaying orthogonal subsets

AbstractCovering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation

Entropy-based convergence rates of greedy algorithms

We present convergence estimates of two types of greedy algorithms in terms of the metric entropy of underlying compact sets. In the first part, we measure the error of a standard greedy reduced basis method for parametric PDEs by the metric entropy of the solution manifold in Banach spaces. This contrasts with the classical analysis based on the Kolmogorov n-widths and enables us to obtain direct comparisons between the greedy algorithm error and the entropy numbers, where the multiplicative constants are explicit and simple. The entropy-based convergence estimate is sharp and improves upon the classical width-based analysis of reduced basis methods for elliptic model problems. In the second part, we derive a novel and simple convergence analysis of the classical orthogonal greedy algorithm for nonlinear dictionary approximation using the metric entropy of the symmetric convex hull of the dictionary. This also improves upon existing results by giving a direct comparison between the algorithm error and the metric entropy.Comment: 22 pages, no figure