We study low-rank tensor-product B-spline (TPBS) models for regression tasks and investigate Dirichlet energy as a measure of smoothness. We show that TPBS models admit a closed-form expression for the Dirichlet energy, and reveal scenarios where perfect interpolation is possible with exponentially small Dirichlet energy. This renders global Dirichlet energy-based regularization ineffective. To address this limitation, we propose a novel regularization strategy based on local Dirichlet energies defined on small hypercubes centered at the training points. Leveraging pretrained TPBS models, we also introduce two estimators for inference from incomplete samples. Comparative experiments with neural networks demonstrate that TPBS models outperform neural networks in the overfitting regime for most datasets, and maintain competitive performance otherwise. Overall, TPBS models exhibit greater robustness to overfitting and consistently benefit from regularization, while neural networks are more sensitive to overfitting and less effective in leveraging regularization.

翻译：本文研究用于回归任务的低秩张量积B样条模型，并探讨狄利克雷能量作为平滑性度量。我们证明张量积B样条模型具有狄利克雷能量的闭式表达式，并揭示了能以指数级小狄利克雷能量实现完美插值的场景。这使得基于全局狄利克雷能量的正则化方法失效。为克服这一局限，我们提出一种基于局部狄利克雷能量的新型正则化策略，该能量定义在以训练点为中心的小超立方体上。借助预训练的张量积B样条模型，我们还针对不完整样本的推断问题提出了两种估计器。与神经网络的对比实验表明：在多数数据集上，张量积B样条模型在过拟合机制中表现优于神经网络，在其他情况下则保持相当性能。总体而言，张量积B样条模型对过拟合具有更强鲁棒性且能持续受益于正则化，而神经网络对过拟合更为敏感且利用正则化的效果较差。