Regularized Regression for Hadamard Manifold-Valued Data: Finite-Sample Bounds and Tangent-Space Sparsity

Regression for Hadamard manifold-valued data requires predictions that account for intrinsic geometry while controlling model complexity. We represent regression curves as Riemannian exponential images of bounded tangent fields and combine an intrinsic loss with penalties on their coefficients or groups. For convex tangent models containing the ambient risk minimizer, we derive finite-sample oracle inequalities under geodesic strong convexity, Lipschitz continuity, a small-ball condition and localized complexity control. The bounds connect tangent-space error, integrated geodesic error and excess risk through an explicit curvature factor. Decomposable penalties express the regularization contribution in terms of subspace compatibility and approximation error, with a sparsity factor for exactly sparse representations. Simulations on the Poincar\'e ball and symmetric positive-definite matrices show a sparsity–prediction trade-off whose benefit diminishes for dense targets. An exploratory EEG covariance-interpolation study on $109$ subjects gives lower mean held-out error than unpenalized fits, while constant and intrinsic regression baselines attain similar errors. These results identify the geometric and statistical conditions supporting tangent regularization and distinguish them from its observed numerical benefits.