Geometric Length Regularization for Robust 2D Data Fitting: Analytical Penalty Bounds and Piecewise-Linear Optimality

We study the problem of approximating a set of two-dimensional points. Classical regularization techniques such as Ridge and Lasso do not provide an explicit upper bound for the penalty coefficient beyond which the solution becomes trivial. We propose a regularization method that penalizes the total arc length of the fitting curve. The optimal solution is shown to be piecewise linear. As the penalty coefficient tends to infinity, the solution degenerates to a straight line – specifically, the average line of the ordinates. Our main result provides an explicit upper bound for the penalty coefficient in terms of the range of the observed ordinates and a prescribed tolerance. For any penalty coefficient below this bound, the optimal solution is guaranteed to deviate from the average line by more than the chosen tolerance, i.e., the solution is non-trivial. Numerical examples, including a sensitivity analysis and comparisons with Ridge and Lasso, illustrate the theoretical findings.