Adaptive Generalized P-Splines for Functional Data: A Statistical Framework via Blockwise GSVD

In this work, we introduce a novel approach for functional data approximation based on generalized P-splines with non-uniform and adaptively placed knots. The key innovation of our proposal is the integration of a conditioning-aware strategy for selecting both the number and positions of the knots, as well as the regularization parameters. By reformulating the Tikhonov regularization problem, we propose a computationally efficient criterion that controls model complexity while ensuring numerical stability. The resulting approximation framework not only improves the fit across the entire functional domain but also maintains compactness and robustness. Extensive numerical experiments conducted on both synthetic and real-world datasets demonstrate that our approach significantly outperforms traditional free knot and smoothing spline methods in terms of approximation error and conditioning.