用离散观测的函数数据做分布式学习，提升回归精度与泛化能力。
Distributed Learning with Discretely Observed Functional Data
- 基于Sobolev核的谱算法，处理离散采样函数数据的分布式回归。
- 在Sobolev范数下证明收敛率上下界匹配，验证条件合理性。
- 适用于高维函数数据建模，适合分布式机器学习场景研究者。
通过选择不同的滤波函数，谱算法可在样本学习框架内生成多种正则化方法以解决统计反问题。本文将分布式谱算法与Sobolev核结合，用于处理函数线性回归问题。算法的设计与数学分析仅要求函数协变量在离散采样点上可观测。算法的假设函数空间由Sobolev核生成，兼具良好的逼近能力和灵活性。通过建立目标函数与函数协变量的正则性条件，本文推导出分布式谱算法在Sobolev范数下的匹配上下界收敛率。这表明所提正则性条件合理，且收敛分析紧致，充分捕捉了函数线性回归的本质特征。文中发展的分析技术与估计方法也改进了已有文献结果。
原文摘要 · Abstract (English)
By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms with Sobolev kernels to tackle the functional linear regression problem. The design and mathematical analysis of the algorithms require only that the functional covariates are observed at discrete sample points. Furthermore, the hypothesis function spaces of the algorithms are the Sobolev spaces generated by the Sobolev kernels, optimizing both approximation capability and flexibility. Through the establishment of regularity conditions for the target function and functional covariate, we derive matching upper and lower bounds for the convergence of the distributed spectral algorithms in the Sobolev norm. This demonstrates that the proposed regularity conditions are reasonable and that the convergence analysis under these conditions is tight, capturing the essential characteristics of functional linear regression. The analytical techniques and estimates developed in this paper also enhance existing results in the previous literature.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。