未知相依设计下的极小极大加性回归

中文摘要 AI 辅助

我们研究了在[0,1]^d上可能非乘积随机设计下的加性回归问题，允许维度d随样本量n增长。我们引入了耦合光滑类，分别控制边际密度和密度加权加性分量的正则性。为处理相依性，我们改编了用于函数型ANOVA模型的Riesz基构造，并在联合密度的均匀界限下建立了与维度无关的常数相容性界。我们构造了阈值化最小二乘估计量，并在适当的维度增长条件下，针对已知或未知边际密度的预测问题建立了匹配的极小极大上界和下界。当边际密度至少与加权分量一样光滑时，未知密度问题达到已知密度的极小极大速率。当密度光滑度较低时，其正则性决定了耦合类上的极小极大速率。最后，我们证明了中心化加性分量可以以相同的聚合上界速率恢复，而无需额外的误差阶。

英文摘要

We study additive regression under an unknown and potentially non product design distribution, allowing the number of covariates to grow with the sample size. We consider a coupled class that separately controls the smoothness of the marginal densities and of each additive component multiplied by the corresponding marginal density. Under joint-density bounds that hold uniformly in the dimension and suitable dimension-growth conditions, we establish matching minimax bounds for prediction. With known marginal densities, the classical additive rate is attainable. When the marginals are unknown, this rate is preserved if the densities are at least as smooth as the weighted components. Otherwise, marginal-density smoothness determines the minimax rate over the coupled class. Finally, we recover all additive components with total squared error of the same order as the prediction error.

发表机构

• Université de Toulouse（图卢兹大学）
• EDF R&D（法国电力公司研发部）
• ANITI（图卢兹人工智能与自然智能研究所）

机构由 AI 辅助整理，请以论文原文为准。