Error Bounds for a Kernel-Based Constrained Optimal Smoothing Approximation

This paper establishes error bounds for the convergence of a piecewise linear approximation of the constrained optimal smoothing problem posed in a reproducing kernel Hilbert space (RKHS). This problem can be reformulated as a Bayesian estimation problem involving a Gaussian process related to the kernel of the RKHS. Consequently, error bounds can be interpreted as a quantification of the maximum a posteriori (MAP) accuracy. To our knowledge, no error bounds have been proposed for this type of problem so far. The convergence results are provided as a function of the grid size, the regularity of the kernel, and the distance from the kernel interpolant of the approximation to the set of constraints. Inspired by the MaxMod algorithm from recent literature, which sequentially allocates knots for the piecewise linear approximation, we conduct our analysis for non-equispaced knots. These knots are even allowed to be non-dense, which impacts the definition of the optimal smoothing solution and our error bound quantifiers. Finally, we illustrate our theorems through several numerical experiments involving constraints such as boundedness and monotonicity.