AI 中文总结
本研究将Johnstone等人的小波稀疏化融入Fan等人的分布式PCA框架，提出的方法在高维下估计误差更优且通信成本更低。
AI 中文摘要
海量数据及数据隐私问题推动了分布式数据技术（亦称为联邦学习）的发展。该场景下，数据子样本被分配至不同机器，需在无法直接访问完整样本的情况下计算统计量。Johnstone与Lu（2009，发表于《美国统计协会期刊》JASA）指出，主成分分析（PCA）在高维情形下统计不一致，并提出通过基于小波的稀疏化与变量选择恢复一致性的方法；Fan等人（2019，发表于《统计学年鉴》AoS）提出了一种估计特征空间的方法——即使数据实际分布式存储，仍可估计合并所有数据后得到的特征空间，但未解决高维情形问题。本研究将Johnstone与Lu（2009）的基于小波的稀疏化融入Fan等人（2019）的分布式PCA框架，旨在降低通信成本且不损害特征空间估计质量。对维度d∈[52,5000]的仿真实验显示：当λ=25时，在d≥152的维度阈值后，所提方法的估计误差优于Fan等人（2019）；当λ=50时，对应阈值为d≥252，且在所研究的全部维度范围内，所提方法传输的系数均显著更少。本研究受巴西圣保罗研究基金会（FAPESP）资助，项目编号#2023/02538-0与#2025/21250-2。
英文摘要
The large volume of data and concerns about data privacy have motivated the development of techniques for distributed data, a problem also known as federated learning. In this scenario, sub-samples of the data are divided across different machines, and statistics must be computed over that data without direct access to the full sample. Johnstone & Lu (2009, JASA) show that principal component analysis (PCA) is statistically inconsistent in the high-dimensional regime, and propose a way to recover consistency through wavelet-based sparsification and variable selection. Fan et al. (2019, AoS) show a way to perform this same estimation -- specifically, to estimate the eigenspace that would be obtained if all the data were pooled together, even though it remains effectively distributed -- without addressing the high-dimensional regime. This work incorporates the wavelet-based sparsification of Johnstone & Lu (2009) into the distributed PCA framework of Fan et al. (2019), aiming to reduce communication cost without compromising the quality of the eigenspace estimation. Simulations across $d \in [52, 5000]$ show that the proposed method overtakes Fan et al. (2019) in estimation error beyond a clear dimensional threshold ($d \geq 152$ for $λ=25$, $d \geq 252$ for $λ=50$), while transmitting systematically fewer coefficients throughout the entire range studied. This study was financed by the Sao Paulo Research Foundation (FAPESP), Brazil. Process Number #2023/02538-0 and Number #2025/21250-2.
Comments10 pages, 3 figures, 1 table
