Limits of PCA
Attempts to fit a hyperplane to the data
•can be interpreted as fitting a Gaussian, where A is the covariance matrix
•this is not a good model for some data
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If you know the model in advance, don’t use PCA
•regression techniques to fit parameters of a model
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Several alternatives/improvements to PCA have been developed
•LLE:  http://www.cs.toronto.edu/~roweis/lle/ 
•isomap:  http://isomap.stanford.edu/ 
•kernel PCA:  http://www.cs.ucsd.edu/classes/fa01/cse291/kernelPCA_article.pdf 
•For a survey of such methods applied to object recognition
–Moghaddam, B., "Principal Manifolds and Probabilistic Subspaces for Visual Recognition", IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), June 2002 (Vol 24, Issue 6, pps 780-788)
http://www.merl.com/papers/TR2002-13/