On the use of the principle of maximum entropy to improve the robustness of bivariate spline least-squares approximation

We consider fitting a bivariate spline regression model to data using a weighted least-squares cost function, with weights that sum to one to form a discrete probability distribution. By applying the principle of maximum entropy, the weight distribution is determined by maximizing the associated entropy function. This approach, previously applied successfully to polynomials and spline curves, enhances the robustness of the regression model by automatically detecting and down-weighting anomalous data during the fitting process. To demonstrate the effectiveness of the method, we present applications to two image processing problems and further illustrate its potential through two synthetic examples. Unlike the standard ordinary least-squares method, the maximum entropy formulation leads to a nonlinear algebraic system whose solvability requires careful theoretical analysis. We provide preliminary results in this direction and discuss the computational implications of solving the associated constrained optimization problem, which calls for dedicated iterative algorithms. These aspects suggest natural directions for further research on both the theoretical and algorithmic fronts.