Estimation of Piecewise Continuous Regression Function in Finite Dimension using Oblique-axis Regression Tree with Applications in Image Denoising

2025-03-20

Statistica sinica

AI Summary

This paper addresses high-precision estimation of piecewise continuous regression functions with complex jump-location curves (JLCs) under fixed finite-dimensional designs. We propose a recursive space-partitioning method based on oblique random trees (ORTs), where local averaging is applied at leaf nodes—marking the first use of ORTs for modeling non-axis-aligned, multi-directional JLCs. Unlike conventional decision trees, which are limited to piecewise constant or smooth functions, our approach significantly enhances preservation of intricate edge structures. Theoretical analysis establishes consistency and convergence rates. Image denoising experiments demonstrate that the method effectively suppresses noise while faithfully retaining jump discontinuities along arbitrary orientations. The core contribution is a novel ORT-based modeling paradigm specifically designed for JLCs, achieving both statistical accuracy and geometric fidelity in representing discontinuous boundaries.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Decision trees are one of the most widely used nonparametric method 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. Given that a two-dimensional grayscale image can be represented as a piecewise-continuous regression function, we apply the proposed algorithm to remove noise from noisy images. Theoretical analysis and numerical results, particularly with image intensity functions, indicate that the proposed method effectively preserves complicated edge structures while efficiently removing noise from piecewise continuous regression surfaces.

Problem

• Estimating piecewise continuous regression functions with complex jump curves
• Preserving intricate edge structures in image denoising applications
• Improving regression tree flexibility for finite-dimensional function estimation

Innovation

• Oblique-axis Regression Tree for piecewise continuous functions
• Clustering pixel intensities via recursive tree partitioning
• Local leaf-only averaging to preserve edge structures