We revisit the classical kernel method of approximation/interpolation theory in a very specific context from the particular point of view of partial differential equations. The goal is to highlight the role of regularization by casting it in terms of actual smoothness of the interpolant obtained by the procedure. The latter will be merely continuous on the data set but smooth otherwise. While the method obtained fits into the category of RKHS methods and hence shares their main features, it explicitly uses smoothness, via a dimension dependent (pseudo-)differential operator, to obtain a flexible and robust interpolant, which can adapt to the shape of the data while quickly transitioning away from it and maintaining continuous dependence on them. The latter means that a perturbation or pollution of the data set, small in size, leads to comparable results in classification applications. The method is applied to both low dimensional examples and a standard high dimensioal benchmark problem (MNIST digit classification).

中文翻译：

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与维度无关的数据集近似及其分类应用

我们从偏微分方程的特定角度，在非常具体的背景下重新审视近似/插值理论的经典核方法。目标是通过根据该过程获得的插值的实际平滑度来投射正则化的作用，从而突出正则化的作用。后者仅在数据集上是连续的，但在其他方面是平滑的。虽然所获得的方法属于 RKHS 方法的类别，因此具有它们的主要特征，但它通过维度相关的（伪）微分算子明确使用平滑度，以获得灵活且鲁棒的插值，该插值可以适应数据，同时快速摆脱它并保持对它们的持续依赖。后者意味着小数据集的扰动或污染会在分类应用中产生可比较的结果。该方法适用于低维示例和标准高维基准问题（MNIST 数字分类）。