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TITLE: [AI] Statistical non-Euclidean data analysis: An introduction and some developments

ABSTRACT: Non-Euclidean data analysis has been an important topic in data science due to rapidly emerging non-Euclidean data such as functional data and manifold-valued data. Analyzing such data is more challenging than analyzing Euclidean data due to the infinite-dimensionality or special geometric structures of non-Euclidean data domains. In fact, the methods for Euclidean data tend to produce inaccurate estimation or prediction when they are applied to non-Euclidean data. We introduce several types of non-Euclidean data including sphere-valued data, shape-valued data, matrix-valued data having some geometric structures and density-valued data. Each data type is introduced with specific real data examples. We also introduce several statistical methods on density estimation and regression analysis for analyzing such data.