Shape constraints offer compelling advantages in nonparametric regression by enabling the estimation of regression functions under realistic assumptions, devoid of tuning parameters. However, most existing shape-constrained nonparametric regression methods, except additive models, impose too few restrictions on the regression functions. This often leads to suboptimal performance, such as overfitting, in multivariate contexts due to the curse of dimensionality. On the other hand, additive shape-constrained models are sometimes too restrictive because they fail to capture interactions among the covariates. In this paper, we introduce a novel approach for multivariate shape-constrained nonparametric regression, which allows interactions without suffering from the curse of dimensionality. Our approach is based on the notion of total concavity originally due to T. Popoviciu and recently described in Gal [24]. We discuss the characterization and computation of the least squares estimator over the class of totally concave functions and derive rates of convergence under standard assumptions. The rates of convergence depend on the number of covariates only logarithmically, and the estimator, therefore, is guaranteed to avoid the curse of dimensionality to some extent. We demonstrate that total concavity can be justified for many real-world examples and validate the efficacy of our approach through empirical studies on various real-world datasets.

翻译：形状约束在非参数回归中提供了引人注目的优势，它能够在无需调节参数的条件下，基于现实假设估计回归函数。然而，除加法模型外，大多数现有的形状约束非参数回归方法对回归函数施加的限制过少。这通常会导致在多元情境下，由于维数灾难而出现次优性能，例如过拟合。另一方面，加法形状约束模型有时又过于严格，因为它们无法捕捉协变量之间的交互作用。本文提出了一种新颖的多元形状约束非参数回归方法，该方法允许交互作用，同时避免了维数灾难。我们的方法基于最初由T. Popoviciu提出、近期在Gal [24]中描述的总凹性概念。我们讨论了在全凹函数类上最小二乘估计量的表征与计算，并在标准假设下推导了其收敛速率。收敛速率仅与协变量数量呈对数依赖关系，因此该估计量在一定程度上保证了避免维数灾难。我们证明总凹性在许多现实案例中具有合理性，并通过在多个真实世界数据集上的实证研究验证了该方法的有效性。