Local-to-global topological methods in data analysis and applications to fMRI of human brain

Pattern recognition and feature extraction tasks in massive and noisy data sets are often plagued by a formidable amount of necessary computations. A common remedy is the application of techniques such as PCA to reduce the dimensionality of data and therefore the amount of computation. While PCA is a useful and popular method, its strict linearity drastically limits its domain of applicability. A number of approaches such as nonlinear formulations of PCA as well as alternatives such as independent component analysis has been proposed. We propose non-linear methods that generalize the PCA and ICA, and lead to the same answer for the cases in which the nonlinearity in the data set is negligible. The main contribution of our research is to develop the nonlinear algorithms based on the geometric ideas in the theory of Riemannian Foliations in differential topology.