Dynamics and robustness for singular value decomposition: application to face recognition
Semmi Pasha, Hamid R. Eghbalnia, Amir H. Assadi · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2001
Principal and Independent Component Analysis (PCA and ICA) are popular and powerful methods for approximation, regression, blind source separation and numerous other statistical tasks. These methods have inherent linearity assumptions that limit their applicability to globally estimate massive and realistic data sets in terms of a few parameters. Global description of such data sets requires more versatile nonlinear methods. Nonetheless, modification of PCA and ICA can be used in a variety of circumstances to discover the underlying non-linear features of the data set. Differential topology and Riemannian geometry have developed systematic methods for local-to-global integration of linearizable features. Numerical methods from approximation theory are applicable to provide a discrete and algorithmic adaptation of continuous topological methods. Such nonlinear descriptions have a far smaller number of parameters than the dimension of the feature space. In addition, it is possible to describe nonlinear relationship between such parameters. We present the mathematical framework for the extension of these methods to a robust estimate for non-linear PCA. We discuss the application of this technique to the study of the topology of the space of parameters in human image databases.