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In this paper we study the statistical properties of Principal ComponentsRegression with Laplacian Eigenmaps (PCR-LE), a method for nonparametricregression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting avector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspacespanned by certain eigenvectors of a neighborhood graph Laplacian. We show thatPCR-LE achieves minimax rates of convergence for random design regression overSobolev spaces. Under sufficient smoothness conditions on the design density$p$, PCR-LE achieves the optimal rates for both estimation (where the optimalrate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) andgoodness-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 designis supported on a manifold of small intrinsic dimension $m$, and give upperbounds 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 ratesof convergence of graph Laplacian eigenvectors to their population-levellimits; in other words, for this problem regression with estimated featuresappears to be much easier, statistically speaking, than estimating the featuresitself. We support these theoretical results with empirical evidence.

Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs