标题： 惩罚非参数回归的场方程 显示英文标题

标题： The Field Equations of Penalized non-Parametric Regression

摘要： 我们通过变分法的视角来看待惩罚风险。 我们考虑由适应性项（例如均方误差）和基于梯度的惩罚组成的风险。 在确立欧拉-拉格朗日场方程作为寻找仅涉及一阶导数的风险最小值的系统方法之后，我们进一步举例说明这种方法应用于由回归函数梯度的平方l2-范数积分惩罚的均方误差。 该风险的最小值给出为一个二阶非齐次偏微分方程的解，其中非齐次项为在特征条件下的目标变量的条件期望。 我们讨论场方程的性质及其实际意义，这些同样适用于线性模型的经典岭惩罚，并将我们的发现融入现有文献中。 特别是，如果我们考虑特征为确定性且均匀分布的，我们可以恢复用于图像去噪的Rudin-Osher-Fatemi模型。 最后，我们概述了未来研究的几个方向。 显示英文摘要

摘要： We view penalized risks through the lens of the calculus of variations. We consider risks comprised of a fitness-term (e.g. MSE) and a gradient-based penalty. After establishing the Euler-Lagrange field equations as a systematic approach to finding minimizers of risks involving only first derivatives, we proceed to exemplify this approach to the MSE penalized by the integral over the squared l2-norm of the gradient of the regression function. The minimizer of this risk is given as the solution to a second order inhomogeneous PDE, where the inhomogeneity is given as the conditional expectation of the target variable conditioned on the features. We discuss properties of the field equations and practical implications thereof, which also apply to the classical Ridge penalty for linear models, and embed our findings into the existing literature. In particular, we find that we can recover the Rudin-Osher-Fatemi model for image-denoising, if we consider the features as deterministic and evenly distributed. Last, we outline several directions for future research.