Principal Component Analysis (PCA)
is one of the most popular techniques for dimensionality reduction of multivariate
with application areas covering many branches of science. The fundamental technique has been extended in a variety of ways including non-linear variants of PCA, combination of local linear PCA, mixture models for PCA, and probabilistic models for PCA. However, conventional PCA handles the multivariate data in a discrete manner only, i.e., the covariance matrix represents only sample data points rather than higher-order data representations.
In this paper we extend conventional
PCA by proposing techniques for constructing the covariance matrix of uniformly
sampled continuous regions in parameter space. These regions include polytops
defined by convex combinations of sample data, and polyhedral regions defined
by the intersection of half spaces. The application of these ideas in practice
are simple and shown to be very effective in providing much superior generalization
properties than conventional PCA for appearance-based recognition applications.