# Convex Formulation for Kernel PCA and its Use in Semi-Supervised Learning

arXiv: https://arxiv.org/abs/1610.06811 · HTML: https://mlsanity.com/paper/1610.06811/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2016.

## Built on by (most cited first, 7 of 7)

- [Multi-view Kernel PCA for Time series Forecasting](https://mlsanity.com/paper/2301.09811/) (2023)
- [Towards Unbiased Random Features with Lower Variance For Stationary Indefinite Kernels](https://mlsanity.com/paper/2104.06204/) (2021)
- [The Lean Data Scientist: Recent Advances towards Overcoming the Data Bottleneck](https://mlsanity.com/paper/2211.07959/) (2022)
- [Duality in Multi-View Restricted Kernel Machines](https://mlsanity.com/paper/2305.17251/) (2023)
- [Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning](https://mlsanity.com/paper/2601.18080/) (2026)
- [A Dual Formulation for Probabilistic Principal Component Analysis](https://mlsanity.com/paper/2307.10078/) (2023)
- [Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA](https://mlsanity.com/paper/2607.21823/) (2026)

## Abstract (arXiv)

In this paper, Kernel PCA is reinterpreted as the solution to a convex optimization problem. Actually, there is a constrained convex problem for each principal component, so that the constraints guarantee that the principal component is indeed a solution, and not a mere saddle point. Although these insights do not imply any algorithmic improvement, they can be used to further understand the method, formulate possible extensions and properly address them. As an example, a new convex optimization problem for semi-supervised classification is proposed, which seems particularly well-suited whenever the number of known labels is small. Our formulation resembles a Least Squares SVM problem with a regularization parameter multiplied by a negative sign, combined with a variational principle for Kernel PCA. Our primal optimization principle for semi-supervised learning is solved in terms of the Lagrange multipliers. Numerical experiments in several classification tasks illustrate the performance of the proposed model in problems with only a few labeled data.