Distributed Tensor Principal Component Analysis with Data Heterogeneity

TL;DR

This work tackles distributed Tensor PCA for $J$-mode tensors stored across multiple sites, where direct pooling is impractical. It introduces three scenarios—homogeneous, heterogeneous with shared components, and transfer—to exploit shared structure while maintaining communication efficiency. The authors develop local-global estimation algorithms, derive sharp statistical guarantees and minimax-rate results, and provide distributed inference for confidence regions. They further extend to knowledge transfer to boost a target site's estimation, with data-adaptive weighting, and validate the methods through simulations and real-data experiments (protein graphs). Overall, the paper advances statistically guaranteed, communication-efficient distributed tensor analysis under data heterogeneity and transfer learning considerations, with practical implications for large-scale, decentralized data analysis.

Abstract

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.

Figures (6)

• Figure 1: The estimation errors of different methods under the homogeneous setting.
• Figure 2: Estimation errors under heterogeneous settings where tensors share the same core.
• Figure 3: Estimation errors under heterogeneous settings where tensors have different cores.
• Figure 4: Comparison of the reconstruction errors within class 0 of the two datasets.
• Figure 5: The coverage rates of different methods under the heterogeneous setting.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• proof : Proof for Theorem 2.1
• Lemma D.1
• Lemma D.2
• proof : Proof of Theorem \ref{['thm:second-lower']}
• proof : Proof of Theorem 3.1
• proof : Proof for Theorem 3.2