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

形状、图像和RNA数据类型在医学、科学、技术等许多领域产生了重要影响，这些现代数据类型包含不同类型的信息，通常不属于欧几里得空间。因此，这些数据类型的分析取决于统计方法的发展，适用于解决非欧几里德度量对象空间的拓扑结构和几何形状。该项目考虑几何拓扑知情的统计方法，分析数据躺在这样的对象空间。开发的方法可以应用于新型图像分析，这些分析更有可能检测到重要特征，识别形状，图像和RNA数据的新位置测量，并提高外围计算机视觉和3D图像数据的质量。所开发的RNA测序数据的方法可以应用于在基因组水平上对病毒性疾病和癌症的研究。该项目还为研究生提供了研究培训机会。该项目开发了物体空间中的统计参数及其推断，这些参数是投影形状数据、数字图像数据和RNA序列分析的基础。该开发依赖于两个关键特征-非线性和紧凑性。从这些数据类型中提取的信息通常表示为分层空间上的点。该项目通过RGB相关性和3D场景重建将经典统计方法扩展到彩色图像，并将处理后的图像视为对象空间上的点，该对象空间可嵌入希尔伯特空间。此外，该项目以提高实际应用所需算法的计算速度为目标，对现有方法进行了扩展，并为这些数据类型引入了免分发方法。该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.