Statistical Analysis of High Dimensional Manifold Data

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Manifold-valued data appear frequently in shape and image analysis, computer vision, biomechanics and many others. Medical imaging data in studies of the variability of human organ shapes lie on a fairly high dimensional nonlinear manifolds, where the challenge is two-fold: high dimensionality with a low sample size (data are expensive to gather) and naturally imposed non-Euclidean geometry. The proposed research aims to answer scientific questions arising in object shape analysis and to provide solid mathematical basis for more complex problems. First, the investigator takes and extends the strategy of backward dimension reduction with regularization framework. Sparse representation of shape and directional data are proposed and their properties are studied. In the regression context, polynomial regression for manifold-valued response to model and test non-geodesic trends is proposed. An extension of local polynomial modeling is also considered. The investigator also explores efficient computational methods. The study of object shape is crucial for understanding the population of human anatomical objects and revealing the interplay between biomarkers/clinical outcomes and object shape variations. Due to the advanced technology, the modern object shape data become big and complex, but conventional methods lack considerations on the special geometric structure of the data types. This research project aims to provide new statistical methodologies for exploratory and confirmatory analysis of the large-scale non-standard data types, including the object shapes. The project will also produce statistical tools, which may be applied in many fields including biomechanics, computer vision, medical studies and biological sciences.