The shattering dimension of sets of linear functionals

• 成果类型: Article
• 署名作者: Mendelson, S; Schechtman, G
• 署名单位: Australian National University; Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000388
• 发表日期: 2004
• 页码: 1746-1770
• 关键词: inequality

摘要

We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization techniques and tail estimates of norms, viewed as random variables on Euclidean spheres. The estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions, a fact which was the original motivation of this study. Although this can probably be done in a more traditional manner, we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem.