# Kernel manifolds: nonlinear-augmentation dimensionality reduction using reproducing kernel Hilbert spaces

## TL;DR

This paper introduces a kernel-based nonlinear augmentation method for dimensionality reduction that leverages reproducing kernel Hilbert spaces to improve approximation accuracy with low training cost.

## Contribution

It generalizes quadratic manifold methods by learning an optimal nonlinear correction in a reproducing kernel Hilbert space, allowing flexible nonlinear structures and improved accuracy.

## Key findings

- Monotonically decreasing error with increasing latent space dimension
- Lower training cost compared to existing methods
- Outperforms proper orthogonal decomposition and recent quadratic manifold approaches

## Abstract

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least-squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. We compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.