| In modern data analysis, we frequently need to analyze objects such as directional data, special types of matrices, probability distributions, and so on. Such structured data are becoming increasingly common in various disciplines. It turns out that many of these data lie on manifolds, which are a natural generalization of Euclidean spaces. The geometry of such a data space (and resulting model space) is crucial to develop more accurate and effective learning models especially when the data space does not exhibit Euclidean geometry. The key focus of this dissertation is to develop statistical machine learning algorithms for the structured data motivated by applications in vision and neuroimaging. The thesis is motivated by some distinct demands of structured data analysis applications covering several scientific domains: 1) How can we model "structured" data in a way that respects the underlying geometry of the data spaces? 2) How can we estimate such models with structured parameters efficiently without leaving the structured data/model spaces? 3) How can we improve the statistical power of statistical machine learning models in cross-sectional and longitudinal analysis that involve structured data spaces?;Using geometrical reasoning, this thesis provides effective statistical learning models for structured data in the context of interpolation, dimensionality reduction and parametric/nonparametric regression for cross-sectional and longitudinal analysis and demonstrates their effectiveness on a broad range of problems motivated from neuroimaging. |
