Abstract
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.
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Acknowledgments
The author is sincerely thankful to the anonymous referee whose comments led to an improvement of the paper.
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Communicated by Wolfgang Dahmen.
Supported by NSF Grant DMS-1309356.
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Guntuboyina, A. Covering Numbers of \(L_{p}\)-Balls of Convex Functions and Sets. Constr Approx 43, 135–151 (2016). https://doi.org/10.1007/s00365-015-9279-1
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DOI: https://doi.org/10.1007/s00365-015-9279-1
Keywords
- Covering numbers
- Packing numbers
- Convex functions
- Integral constraints
- Metric entropy
- Komogorov \(\epsilon \)-entropy