Totally Concave Regression

• 成果类型: Article; Early Access
• 署名作者: Ki, Dohyeong; Guntuboyina, Adityanand
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2026.2620145
• 发表日期: 2026-03-23
• 关键词: Interaction effect modeling; Mixed partial derivative; multivariate convex regression; Popoviciu's convex function; Shape-constrained estimation; least-squares estimation; convex regression; isotonic regression; convergence-rates; risk bounds; shape; Consistency; models; MONOTONICITY; estimators

摘要

Shape constraints in nonparametric regression provide a powerful framework for estimating regression functions under realistic assumptions without tuning parameters. However, most existing methods-except additive models-impose too weak restrictions, often leading to overfitting in high dimensions. Conversely, additive models can be too rigid, failing to capture covariate interactions. This article introduces a novel multivariate shape-constrained regression approach based on total concavity, originally studied by T. Popoviciu. Our method allows interactions while mitigating the curse of dimensionality, with convergence rates that depend only logarithmically on the number of covariates. We characterize and compute the least squares estimator over totally concave functions, derive theoretical guarantees, and demonstrate its practical effectiveness through empirical studies on real-world datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.