Kernel-based framework for spectral dimensionality reduction and clustering formulation: A theoretical study

This work outlines a unified formulation to represent spectral approaches for both dimensionality reduction and clustering. Proposed formulation starts with a generic latent variable model in terms of the projected input data matrix. Particularly, such a projection maps data onto a unknown high-dimensional space. Regarding this model, a generalized optimization problem is stated using quadratic formulations and a least-squares support vector machine. The solution of the optimization is addressed through a primal-dual scheme. Once latent variables and parameters are determined, the resultant model outputs a versatile projected matrix able to represent data in a low-dimensional space, as well as to provide information about clusters. Particularly, proposed formulation yields solutions for kernel spectral clustering and weighted-kernel principal component analysis.