In a high-dimensional regression framework, we study consequences of the naive two-step procedure where first the dimension of the input variables is reduced and second, the reduced input variables are used to predict the output variable with kernel regression. In order to analyze the resulting regression errors, a novel stability result for kernel regression with respect to the Wasserstein distance is derived. This allows us to bound errors that occur when perturbed input data is used to fit the regression function. We apply the general stability result to principal component analysis (PCA). Exploiting known estimates from the literature on both principal component analysis and kernel regression, we deduce convergence rates for the two-step procedure. The latter turns out to be particularly useful in a semi-supervised setting.

Locally optimal Delaunay triangulations are constructed to improve previous image approximation schemes. Our construction relies on a local optimization procedure, termed exchange. The efficient implementation of the exchange algorithm is addressed, and its complexity is discussed. The good performance of our improved image approximation is illustrated by numerical comparisons.