# A kernel Principal Component Analysis (kPCA) digest with a new backward mapping (pre-image reconstruction) strategy

arXiv: https://arxiv.org/abs/2001.01958 · HTML: https://mlsanity.com/paper/2001.01958/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2020.

## Built on by (most cited first, 4 of 4)

- [High Dimensional Bayesian Optimization with Kernel Principal Component Analysis](https://mlsanity.com/paper/2204.13753/) (2022)
- [Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reduction](https://mlsanity.com/paper/2302.06528/) (2023)
- [Nonintrusive Uncertainty Quantification for automotive crash problems with VPS/Pamcrash](https://mlsanity.com/paper/2102.07673/) (2021)
- [A Matter of Time: Revealing the Structure of Time in Vision-Language Models](https://mlsanity.com/paper/2510.19559/) (2025)

## Abstract (arXiv)

Methodologies for multidimensionality reduction aim at discovering low-dimensional manifolds where data ranges. Principal Component Analysis (PCA) is very effective if data have linear structure. But fails in identifying a possible dimensionality reduction if data belong to a nonlinear low-dimensional manifold. For nonlinear dimensionality reduction, kernel Principal Component Analysis (kPCA) is appreciated because of its simplicity and ease implementation. The paper provides a concise review of PCA and kPCA main ideas, trying to collect in a single document aspects that are often dispersed. Moreover, a strategy to map back the reduced dimension into the original high dimensional space is also devised, based on the minimization of a discrepancy functional.