Adaptive sparse-knot spline models for approximating sequences of noisy data points in arbitrary-dimensional spaces

We address the problem of adaptive approximation of sequences of noisy data points in arbitrary-dimensional spaces using spline models of any degree and smoothness. Knot vectors are determined as solutions of unconstrained sparse optimization problems, yielding a flexible and compact representation based on a limited number of polynomial segments. Unlike previously proposed methods, the resulting knot vectors may include breakpoints with arbitrary multiplicity. Furthermore, the proposed approach enables the reconstruction of both scalar functions and open or closed parametric curves, always relying on a simple, fully unconstrained optimization framework that ensures robustness and computational efficiency.