Efficient Nonparametric Estimation of 3D Point Cloud Signals through Distributed Learning

Advancements in technology have elevated the prominence of 3D point cloud data, making its analysis increasingly vital across various applications. This need drives the demand for advanced statistical analytic approaches to handle challenges such as size, sparsity, and irregularity in 3D point clouds while ensuring accurate and efficient information extraction. This article introduces a novel nonparametric distributed (NPD) learning framework that uses trivariate spline smoothing over a triangulation of domain. The proposed NPD algorithm features a straightforward, scalable, and communication-efficient implementation scheme that can achieve near-linear speedup. In addition, we provide rigorous theoretical support for the NPD estimation framework and demonstrate that the NPD spline estimators attain the same convergence rate as the global spline estimators obtained using the entire dataset and achieve the optimal nonparametric convergence rate established by Stone under some regularity conditions. To evaluate the efficacy of the proposed NPD method, we conduct simulation studies comparing it with several global nonparametric estimation methods used to smooth the 3D data. The results demonstrate the superior performance of the NPD method in accurately and efficiently processing and learning from 3D point clouds, highlighting its potential to advance large and complex data analysis. Supplementary material, which contains related technical details, proofs of the theoretical results, and additional results in simulation studies, is available online.

技术进步提升了三维点云（3D point cloud）数据的重要性，使其分析在诸多应用场景中愈发关键。这一需求催生了对先进统计分析方法的迫切需求，以应对三维点云数据在规模、稀疏性与不规则性等方面的挑战，同时保障信息提取的精准性与高效性。本文提出一种新颖的非参数分布式（nonparametric distributed, NPD）学习框架，该框架基于定义域三角剖分开展三元样条平滑操作。所提出的NPD算法拥有简洁易实现、可扩展性优异且通信高效的实现方案，可实现近乎线性的加速比。此外，本文为NPD估计框架提供了严谨的理论支撑，并证明：在若干正则性条件下，NPD样条估计器可达到与使用全量数据集得到的全局样条估计器一致的收敛速度，同时达成由Stone确立的最优非参数收敛速率。为验证所提NPD方法的有效性，本文开展了仿真实验，将其与若干用于三维数据平滑的全局非参数估计方法进行对比。实验结果表明，NPD方法在三维点云数据的精准高效处理与学习任务中表现更优，凸显了其在推动大规模复杂数据分析发展方面的应用潜力。本文相关技术细节、理论结果证明以及仿真实验补充结果等补充材料均可在线获取。