Decision trees are one of the most widely used nonparametric methods for regression and classification. In existing literature, decision tree-based methods have been used for estimating continuous functions or piecewise-constant functions. However, they are not flexible enough to estimate the complex shapes of jump location curves (JLCs) in two-dimensional regression functions. In this article, we explore the Oblique-axis Regression Tree (ORT) and propose a method to efficiently estimate piece-wise continuous functions in a general finite dimension with fixed design points. The central idea involves clustering the local pixel intensities by recursive tree partitioning and using the local leaf-only averaging for estimation of the regression function at a given pixel. The proposed method can preserve complex shapes of the JLCs well in a finite-dimensional regression function. Due to a different set of assumptions on the underlying regression function, the overall framework of the proofs is different from what is available in the literature on regression trees. Theoretical analysis and numerical results, particularly on image denoising, indicate that the proposed method effectively preserves complicated edge structures while efficiently removing noise from piecewise continuous regression surfaces.

翻译：决策树是回归和分类问题中最广泛使用的非参数方法之一。现有文献中，基于决策树的方法已被用于估计连续函数或分段常数函数。然而，这些方法在估计二维回归函数中跳跃位置曲线复杂形状时灵活性不足。本文探讨了斜轴回归树，并提出一种在固定设计点条件下高效估计一般有限维空间中分段连续函数的方法。其核心思想是通过递归树划分对局部像素强度进行聚类，并利用局部叶节点平均来估计给定像素处的回归函数。所提方法能够很好地保留有限维回归函数中跳跃位置曲线的复杂形状。由于对底层回归函数采用了不同的假设集，证明的整体框架与现有回归树文献中的方法有所不同。理论分析和数值结果（特别是在图像去噪方面）表明，该方法在有效去除分段连续回归表面噪声的同时，能较好地保持复杂的边缘结构。