Modern scientific applications increasingly generate data that take values in non-Euclidean spaces, including covariance matrices in neuroimaging, probability distributions in single-cell genomics, and correlation structures in network analysis. Classical regression methods, designed for Euclidean responses, fail to account for the geometric constraints inherent in these data types, motivating the development of new theoretical and computational frameworks. This thesis focuses on regression analysis for covariance matrix-valued responses within the framework of Fr'echet regression, which generalizes linear regression to arbitrary metric spaces by relying solely on distance rather than linear structure. We provide the first comprehensive statistical theory and methodology for Fr'echet regression in a metric space with non-zero curvature. Our theoretical contributions include establishing non-asymptotic uniform consistency and asymptotic normality for the Fr'echet regression estimator. For inference, we develop two novel testing procedures: (i) a global effects test with an asymptotic null distribution following a weighted sum of chi-squares, where weights derive from the covariance structure, and (ii) a partial effects test based on reproducing kernel Hilbert space (RKHS) methodology, also yielding a weighted chi-square limit.

We validate our methods through extensive simulations and apply them to single-cell gene co-expression networks, where we uncover significant age-related changes in nutrient-sensing pathway structures. These contributions establish a rigorous theoretical foundation and provide practical statistical tools that may facilitate future developments for analyzing other types of non-Euclidean data encountered in modern scientific research.