Geometric properties of the ridge function manifold
Geometric properties of the ridge function manifold
复制标题
岭函数流形的几何性质
DOI：
10.1007/s10444-008-9106-3
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发表时间：
2010
影响因子：
1.7
通讯作者：
V. Maiorov
中科院分区：
文献类型：
--
作者：
V. Maiorov
研究了由n个函数的所有可能的线性组合构成的脊函数流形$\mathcal{R}_n$的几何性质，其中a·x是${\mathbb R}^d$中的内积.我们得到了紧类Gn，s在s次多项式空间${\mathbb R}^d $上的ε-覆盖数较小的ε-熵数的一个估计.特别地，我们证明了当n ≤ sd − 1时，类Gn，s在空间Lq中的ε-熵数Hε（Gn，s，Lq）的阶为nsystem 1/ε（模对数因子）。注意单位球的ε-熵数H_\vareps（B\mathcal{P}_s^d，L_q）$的阶为sdlog 1/ε。此外，我们还得到了岭函数类Gn，s的伪维数的估计。
We study geometrical properties of the ridge function manifold $\mathcal{R}_n$ consisting of all possible linear combinations of n functions of the form g(a· x), where a·x is the inner product in ${\mathbb R}^d$. We obtain an estimate for the ε-entropy numbers in terms of smaller ε-covering numbers of the compact class Gn,s formed by the intersection of the class $\mathcal{R}_n$ with the unit ball $B\mathcal{P}_s^d$ in the space of polynomials on ${\mathbb R}^d$ of degree s. In particular we show that for n ≤ sd − 1 the ε-entropy number Hε(Gn,s,Lq) of the class Gn,s in the space Lq is of order nslog1/ε (modulo a logarithmic factor). Note that the ε-entropy number $H_\varepsilon(B\mathcal{P}_s^d,L_q)$ of the unit ball is of order sdlog1/ε. Moreover, we obtain an estimate for the pseudo-dimension of the ridge function class Gn,s.