Collaborative Research: Advances in the Theory and Practice of Non-Euclidean Statistics

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Shape, image, and RNA data types have had an important impact in many areas of Medicine, Science, Technology, etc. These modern data types contain different kinds of information that typically do not belong to a Euclidean space. As such, the analysis of these data types depends upon the development of statistical methodology that is adapted to address the topology and geometry of non-Euclidean metric object spaces. The project considers geometric-topologically informed statistical methods for the analysis of data lying on such object spaces. The methods to develop can find applications for new kinds of image analyses that are more likely to detect important features, identify new measures of location for shape, image, and RNA data, and improve the quality of peripheral computer vision and 3D image data. The developed methods for RNA sequencing data may apply to the investigation of viral diseases and cancer at the genomic level. The project also provided research training opportunities for graduate students.The project develops statistical parameters and their inference in object spaces that underlie projective shape data, digital image data, and RNA sequence analyses. The development relies upon two key features - nonlinearity and compactness. Information extracted from these data types is often represented as points on a stratified space. This project extends classical statistical methods to colored images via RGB correlations and 3D scene reconstructions and considers the processed images to be points on an object space, which is embeddable in a Hilbert space. Moreover, this project expands upon existing approaches with the aim of increasing the computational speed of algorithms required for real-world applications and introducing distribution-free methodologies for these data types.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.