Learning functions varying along an active subspace.
Learning functions varying along an active subspace.
复制标题
学习函数沿着活动子空间变化。
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Wenjing Liao
中科院分区：
文献类型：
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作者：
Hao Liu;Wenjing Liao
许多感兴趣的函数都在高维空间中，但表现出低维结构。本文研究了一个$S$-Holder函数$f$在$mathbb{R}^D$中的回归，它沿维度为$d$的活动子空间变化，而在$dll D$中变化。以$varepsilon$精度直接逼近$mathbb{R}^D$中的$f$需要按$varepsilon^{-(2S+D)/S}$的顺序的样本数$n$。在本文中，我们改进了广义轮廓回归(GCR)算法来估计活动子空间，并使用分段多项式来逼近函数。GCR是最好的活动子空间估计器之一，但其样本复杂性是一个悬而未决的问题。改进的GCR提高了原GCR的效率，当$n$足够大时，活动子空间的均方估计误差为$O(n^{-1})$。证明了$f$的均方回归误差为$Left(n/logn
其中，指数取决于活动子空间$d$的维度，而不是环境空间$D$。这一结果表明，GCR在学习低维活动子空间方面是有效的。通过多次数值实验验证了算法的收敛速度。
Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This paper studies regression of a $s$-Holder function $f$ in $mathbb{R}^D$ which varies along an active subspace of dimension $d$ while $dll D$. A direct approximation of $f$ in $mathbb{R}^D$ with an $varepsilon$ accuracy requires the number of samples $n$ in the order of $varepsilon^{-(2s+D)/s}$. In this paper, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to an mean squared estimation error of $O(n^{-1})$ for the active subspace, when $n$ is sufficiently large. The mean squared regression error of $f$ is proved to be in the order of $left(n/log n
ight)^{-frac{2s}{2s+d}}$ where the exponent depends on the dimension of the active subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning low-dimensional active subspaces. The convergence rate is validated through several numerical experiments.
影响因子：
1.5
作者：
Lanteri, Alessandro;Maggioni, Mauro;Vigogna, Stefano
通讯作者：
Vigogna, Stefano