Entropy of convex hulls: some Lorentz norm results

Authors:
Ingo Steinwart
Affiliations:
Los Alamos National Loboratary, Computer and Computational Sciences Division, Modeling, Algorithms, and Informatics Group, CCS-3 Machine Learning and Pattern Recongnition, Mail Stop B256, Los Alam ...
Venue:
Journal of Approximation Theory
Year:
2004

Citing 2
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Abstract

Let A be a subset of a type p Banach space E, 1 p ≤ 2, such that its entropy numbers satisfy, (εn(A))n ∈ ℓq,s for some q,s ∈ (0, ∞). We show (en(aco A))n ∈ ℓr,s for the dyadic entropy numbers of the absolutely convex hull aco A of A, where r is defined by 1/r = 1/p' + 1/q. Furthermore, we show for slowly decreasing entropy numbers that (en(A))n ∈ ℓq,s implies (en(aco A))n ∈ ℓp',s for all 0 s q defined by 1/q = 1/p' + 1/s.