Entropy of Convex Functions on ℝ d.
Entropy of Convex Functions on ℝ d.
复制标题
ℝ d 上凸函数的熵。
DOI：
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发表时间：
2015
影响因子：
2.7
通讯作者：
J. Wellner
中科院分区：
文献类型：
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作者：
Fuchang Gao;J. Wellner
设Ω是一个有界的闭凸集，在一个非空的内部，设？r （Ω）是Ω上的一类凸函数，其r -范数以1为界。我们得到了ε-熵的精确估计。p （Ω）指标下r （Ω）， 1≤p < r≤∞。特别地，结果表明所有d-多面体的普遍下界ε-d/2也是上界，并且[公式：见文]的[公式：见文]的普遍上界是由闭合单位球得到的。虽然一般凸体可以用内切多面体近似，但熵率不能传递到极限体。我们的结果应用于高维形状约束函数的非参数估计的收敛速率问题。
Let Ω be a bounded closed convex set in ℝ d with non-empty interior, and let ? r (Ω) be the class of convex functions on Ω with Lr -norm bounded by 1. We obtain sharp estimates of the ε-entropy of ? r (Ω) under Lp (Ω) metrics, 1 ≤ p < r ≤ ∞. In particular, the results imply that the universal lower bound ε-d/2 is also an upper bound for all d-polytopes, and the universal upper bound of [Formula: see text] for [Formula: see text] is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.