6 research outputs found
On the size of convex hulls of small sets
We investigate two different notions of "size" which appear naturally in Statistical Learning
Theory. We present quantitative estimates on the fat-shattering dimension and on the
covering numbers of convex hulls of sets of functions, given the necessary data on the
original sets. The proofs we present are relatively simple since they do not require extensive
background in convex geometry
Risk Bounds for Mixture Density Estimation
In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Procedure (MLE) and the greedy procedure described by Li and Barron. Approximation and estimation bounds are given for the above methods. We extend and improve upon the estimation results of Li and Barron, and in particular prove an bound on the estimation error which does not depend on the number of densities in the estimated combination
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