Riemannian Ordinary Least Squares

Many scientific responses take values in Riemannian manifolds, with predictors that are scalar- and/or manifold-valued. A regression analysis in such settings must support not only point prediction but also formal hypothesis testing and effect-size estimation for individual coefficients. Yet no current manifold-regression framework simultaneously accommodates both predictor types, and returns per-coefficient hypothesis tests. We propose Riemannian Ordinary Least Squares (ROLS), a tangent-space conditional-mean model that serves as the OLS analogue for this setting. ROLS is well defined under an explicit injectivity condition that keeps the logarithm maps single-valued, with a cut-locus diagnostic for branch-sensitive cases outside it. Within the local geometric setting it admits a closed-form estimator, an $O(n^{-1})$ excess prediction-error bound with explicit curvature bias, and asymptotic $t$- and Wald tests for individual coefficients. Across simulations with synthetic data and three case studies with real data, ROLS is competitive with the strongest prediction baselines and identifies scientifically meaningful predictors, providing for manifold-valued regression the coefficient-level inference that OLS provides in the Euclidean case.