Dimensionality Reduction and Wasserstein Stability for Kernel Regression
This work addresses the stability and error analysis of a common high-dimensional regression approach, providing theoretical guarantees for practitioners using dimensionality reduction and kernel regression, but it is incremental as it builds on existing literature for PCA and kernel regression.
The paper tackles the problem of analyzing regression errors in a two-step procedure combining dimensionality reduction and kernel regression, deriving a novel stability result with respect to the Wasserstein distance to bound errors from perturbed input data, and applies it to PCA to deduce convergence rates, particularly useful in semi-supervised settings.
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.