Entropy of Convex Functions on $$mathbb {R}^d$$Rd
Entropy of Convex Functions on $$mathbb {R}^d$$Rd
复制标题
$$mathbb {R}^d$$Rd 上凸函数的熵
DOI：
--
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Wellner
中科院分区：
文献类型：
--
作者：
Fuchang Gao;J. Wellner
令 $$varOmega $$Ω 为 $$mathbb {R}^d$$Rd 中内部非空的有界闭凸集，并令 $${mathcal C}_r(varOmega )$$Cr(Ω) 为 $$varOmega $$Ω 上的凸函数类，$$L^r$$Lr-范数以 1 为界。我们获得 $$varepsilon $$ε-熵的精确估计$${mathcal C}_r(varOmega )$$Cr(Ω) 在 $$L^p(varOmega )$$Lp(Ω) 指标下的 $$1le p<rle infty $$1≤p<r≤∞。特别是，结果表明通用下界 $$varepsilon ^{-d/2}$$ε-d/2 也是所有 d-多胞体的上限，并且 $$varepsilon ^{-frac{(d-1)}{2}cdot frac{pr}{r-p}}$$ε-(d-1)2·prr-p 的通用上限$$p>frac{dr}{d+(d-1)r}$$p>drd+(d-1)r 是通过闭合单位球获得的。虽然一般的凸体可以通过内接多面体来近似，但熵率不会转移到限制体上。我们的结果可应用于有关高维形状约束函数的非参数估计器收敛率的问题。
Let $$varOmega $$Ω be a bounded closed convex set in $$mathbb {R}^d$$Rd with nonempty interior, and let $${mathcal C}_r(varOmega )$$Cr(Ω) be the class of convex functions on $$varOmega $$Ω with $$L^r$$Lr-norm bounded by 1. We obtain sharp estimates of the $$varepsilon $$ε-entropy of $${mathcal C}_r(varOmega )$$Cr(Ω) under $$L^p(varOmega )$$Lp(Ω) metrics, $$1le p<rle infty $$1≤p<r≤∞. In particular, the results imply that the universal lower bound $$varepsilon ^{-d/2}$$ε-d/2 is also an upper bound for all d-polytopes, and the universal upper bound of $$varepsilon ^{-frac{(d-1)}{2}cdot frac{pr}{r-p}}$$ε-(d-1)2·prr-p for $$p>frac{dr}{d+(d-1)r}$$p>drd+(d-1)r 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.