  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.
