# Dimensionality Reduction and Wasserstein Stability for Kernel Regression

arXiv: https://arxiv.org/abs/2203.09347 · HTML: https://mlsanity.com/paper/2203.09347/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 58 references that are in the map.
- First posted on arXiv 17 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Generalization Error Rates in Kernel Regression: The Crossover from the Noiseless to Noisy Regime](https://mlsanity.com/paper/2105.15004/) (2021)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Out-of-Distribution Generalization in Kernel Regression](https://mlsanity.com/paper/2106.02261/) (2021)
- [Total stability of kernel methods](https://mlsanity.com/paper/1709.07625/) (2017)

## Built on by (most cited first, 3 of 3)

- [On the Connection between Lp and Risk Consistency and its Implications on Regularized Kernel Methods](https://mlsanity.com/paper/2303.15210/) (2023)
- [Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2504.13543/) (2025)
- [On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2504.13623/) (2025)

## Abstract (arXiv)

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.