Geometry, Shape and Objects

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Filaments, ridges, graphs, contours, and paths are examples of geometric objects in modern data analytic challenges. The development and the analysis of statistical methodology to tackle challenges related to such objects is the overarching goal of this project. In particular, the principal investigator and his collaborators will (i) develop statistical methodology and algorithms for the estimation of ridge lines and/or filamentary structure involving path smoothing methodologies. Here, paths are estimates of integral curves of gradients or of appropriate eigenvectors of Hessians. (ii) They will also develop dimension reduction methodology for subsequent clustering of high-dimensional data, where projection onto non-linear manifolds will be considered. Further goals are (iii) the adaptation of model selection methodologies for high-dimensional graphical models to incorporate spatial and temporal smoothness, and (iv) the investigation of commonalities and differences between statistical density/regression level set methodology and depth contour approaches. Besides the intrinsic geometric nature of the problems under consideration, geometry also comes into play via the choice of an appropriate distance measure, for instance.This project will lead to new statistical methodologies underpinned by relevant theory, and to corresponding numerical algorithms in the area of geometric statistics. In this field of statistics, the objects of interest itself are genuinely geometrically motivated. One important guiding instance of a relevant scientific problem is the analysis of the cosmic web, which consists of locations of galaxies in the space. The web-like structure of these locations requires the use of nonstandard methodologies. Geometrically motivated objects also play a major role in several other scientific areas, such as medical imaging, fingerprint identification and remote sensing. The findings of this project will advance the field of statistics, and they will directly impact the relevant fields of application. The education of graduate and undergraduate students in a modern field of statistics is another important goal of this project.