As tensors become widespread in modern data analysis, Tucker low-rank Principal Component Analysis (PCA) has become essential for dimensionality reduction and structural discovery in tensor datasets. Motivated by the common scenario where large-scale tensors are distributed across diverse geographic locations, this paper investigates tensor PCA within a distributed framework where direct data pooling is impractical. We offer a comprehensive analysis of three specific scenarios in distributed Tensor PCA: a homogeneous setting in which tensors at various locations are generated from a single noise-affected model; a heterogeneous setting where tensors at different locations come from distinct models but share some principal components, aiming to improve estimation across all locations; and a targeted heterogeneous setting, designed to boost estimation accuracy at a specific location with limited samples by utilizing transferred knowledge from other sites with ample data. We introduce novel estimation methods tailored to each scenario, establish statistical guarantees, and develop distributed inference techniques to construct confidence regions. Our theoretical findings demonstrate that these distributed methods achieve sharp rates of accuracy by efficiently aggregating shared information across different tensors, while maintaining reasonable communication costs. Empirical validation through simulations and real-world data applications highlights the advantages of our approaches, particularly in managing heterogeneous tensor data.

翻译：随着张量在现代数据分析中的广泛应用，Tucker低秩主成分分析（PCA）已成为张量数据集降维和结构发现的关键技术。针对大规模张量分布在异地的常见场景，本文研究了直接数据聚合不可行的分布式框架下的张量PCA问题。我们系统分析了分布式张量PCA中的三类具体场景：同构场景——异地张量均来自同一含噪模型；异构场景——异地张量虽来自不同模型但共享部分主成分，旨在提升所有位置的估计性能；以及目标导向的异构场景——通过利用其他数据丰富站点的迁移知识，提升特定样本稀疏站点的估计精度。我们针对每种场景提出了创新的估计方法，建立了统计保证理论，并开发了用于构建置信区域的分布式推断技术。理论结果表明，这些分布式方法通过高效聚合不同张量间的共享信息，在保持合理通信开销的同时实现了最优收敛速率。通过仿真实验和真实数据应用验证，我们的方法在管理异构张量数据方面展现出显著优势。