  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.
