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.

翻译：本文针对再生核希尔伯特空间（RKHS）中约束最优平滑问题的分段线性逼近收敛性建立了误差界。该问题可重新表述为涉及与RKHS核相关的高斯过程的贝叶斯估计问题，因此误差界可解释为最大后验概率（MAP）估计精度的量化。据我们所知，目前尚未有研究为此类问题提出误差界。收敛结果以网格尺寸、核函数正则性以及逼近的核插值到约束集的距离作为函数给出。受近期文献中MaxMod算法的启发——该算法为分段线性逼近顺序分配节点，我们对非等距节点进行了分析。这些节点甚至允许非稠密分布，这影响了最优平滑解的定义及误差界量化指标。最后，我们通过涉及有界性和单调性等约束的若干数值实验验证了所提定理。