In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.
翻译:在本文中,我们研究了使用 Laplacecian Eigenmaps (PCR-LE) 的主要部件回归的统计属性。 PCR-LE 是使用 Laplacecian Eigenmaps (LE) 的一种非参数回归方法。 Pplacian Eigenmaps (LE) 的计算结果。 PCR-LE 的计算方法是将观测反应的矢量 $\bf Y} = (Y_1,\ldots,Y_n) 投射到由某个周边图形 Laplacian 的某些电子元数覆盖的子空间范围内。 我们显示, PCR- LE 在随机设计回归回归的最小值回归速度方面, 在设计密度 $p p$ 的足够平稳条件下, PCRLE 达到最佳反应量 / m (2) + d} 美元 和 质量测试结果检测结果(n+ $ 美元) 。 我们还显示, PCR- lCR- dalental 的递解算算算算算算算数据本身的数值, 上, 的数值的数值本身的精确值本身的数值是支持的。