Covering Numbers of \(L_{p}\) -Balls of Convex Functions and Sets

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• 作者：Adityanand Guntuboyina
• 关键词：Covering numbers ; Packing numbers ; Convex functions ; Integral constraints ; Metric entropy ; Komogorov $$\epsilon $$ ϵ ; entropy ; 41A46 ; 46B10 ; 52A10 ; 52A41 ; 54C70
• 刊名：Constructive Approximation
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：43
• 期：1
• 页码：135-151
• 全文大小：493 KB
• 参考文献：1.Bronshtein, E.M.: \(\epsilon \) -entropy of convex sets and functions. Sib. Math. J. 17, 393–398 (1976)CrossRef
  2.Dryanov, D.: Kolmogorov entropy for classes of convex functions. Constr. Approx. 30, 137–153 (2009)MATH MathSciNet CrossRef
  3.Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974)MATH MathSciNet CrossRef
  4.Guntuboyina, A., Sen, B.: Covering numbers for convex functions. IEEE Trans. Inf. Theory 59(4), 1957–1965 (2013)MathSciNet CrossRef
  5.Rockafellar, R.T.: Convex Analysis. Princeton Univ. Press, Princeton, NJ (1970)MATH CrossRef
  6.Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge Univ. Press, Cambridge (1993)MATH CrossRef
  
• 作者单位：Adityanand Guntuboyina (1)
  
  1. Department of Statistics, University of California, 423 Evans Hall, Berkeley, CA, 94720, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it with weaker integral constraints. The existing results can be recovered as special cases of our results. Keywords Covering numbers Packing numbers Convex functions Integral constraints Metric entropy Komogorov \(\epsilon \)-entropy