Minimax optimal regression over Sobolev spaces via Laplacian Eigenmaps on neighbourhood graphs

In this paper, we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for non-parametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\textbf Y} = (Y_1,\ldots ,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighbourhood 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 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.