论文标题
学习功能沿中央子空间变化
Learning functions varying along a central subspace
论文作者
论文摘要
许多感兴趣的功能都在高维空间中，但表现出低维结构。本文研究了$ s-hölder函数$ f $ in $ \ mathbb {r}^d $的回归，该$ f $ a $ \ mathbb {r}^d $随着尺寸$ d $的中央子空间而变化，而$ d \ ll d $。 $ \ Mathbb {r}^d $的直接近似与$ \ varepsilon $精度需要$ \ varepsilon^{ - （2S+d）/s} $的样本$ n $的数量。在本文中，我们分析了广义轮廓回归（GCR）算法以估计中央子空间，并使用分段多项式进行函数近似。 GCR是中央子空间的最佳估计量之一，但其样本复杂性是一个悬而未决的问题。我们证明，如果差异数量已知，则GCR导致中央子空间的均值估计误差为$ O（n^{-1}）$。本文还给出了该方差数量的估计误差。事实证明，$ f $的平均平方回归误差的顺序（n/\ log n \ right）^{ - \ frac {2s} {2S} {2S+d}} $，其中指数取决于中央子空间$ d $的尺寸，而不是环境空间$ d $。该结果表明GCR有效地学习了低维中心子空间。我们还提出了一个改进的GCR，其效率提高。通过几个数值实验验证了收敛速率。
Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This paper studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along a central subspace of dimension $d$ while $d\ll D$. 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 analyze the Generalized Contour Regression (GCR) algorithm for the estimation of the central subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the central subspace, but its sample complexity is an open question. We prove that GCR leads to a mean squared estimation error of $O(n^{-1})$ for the central subspace, if a variance quantity is exactly known. The estimation error of this variance quantity is also given in this paper. The mean squared regression error of $f$ is proved to be in the order of $\left(n/\log n\right)^{-\frac{2s}{2s+d}}$ where the exponent depends on the dimension of the central subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning the low-dimensional central subspace. We also propose a modified GCR with improved efficiency. The convergence rate is validated through several numerical experiments.
