This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of indifference. It is shown that if a probability measure on an infinite-dimensional function space possesses natural symmetries (invariance under translation, rotation, scaling, and Gaussianity), then the entire solution scheme, including the kernel form, the type of regularization, and the noise parameterization, follows analytically from these postulates. The resulting kernel coincides with a generalized polyharmonic spline; however, unlike existing approaches, it is not chosen empirically but arises as a consequence of the indifference principle. This result provides a theoretical foundation for a broad class of smoothing and interpolation methods, demonstrating their optimality in the absence of a priori information.

翻译：本文在随机函数理论框架下，将机器学习回归问题作为多元逼近问题进行研究。提出了一种基于无差别性假设的回归方法从头推导。研究表明，若无限维函数空间上的概率测度具有自然对称性（平移、旋转、缩放不变性及高斯性），则整个求解方案——包括核函数形式、正则化类型及噪声参数化——均可从这些假设出发解析导出。所得核函数与广义多调和样条一致；但与现有方法不同，该核函数并非经验选择，而是无差别性原理的必然结果。这一结论为广泛类别的平滑与插值方法提供了理论基础，证明了其在缺乏先验信息条件下的最优性。