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                                   Principal Component Analysis Over 
                    Continuous Subspaces and Intersection of Half-spaces

                                            Amnon Shashua
                                 Hebrew University of Jerusalem

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Principal Component Analysis (PCA) is one of the most popular techniques
for dimensionality reduction of multivariate data points 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.

This work was jointly done with Anat Levin.