In this work, we model the observed multi-view tensors by decomposing the underlying signal in each view into two components: (i) the shared component that captures common dynamics across all views, and (ii) the private component that accounts for view-wise unique variations. To decouple the shared and private components, we introduce a novel Tucker personalized subspace principal component analysis (TPS-PCA) approach for tensors, which admits a one-step closed-form solution and serves as an ideal surrogate for our extended tensor-version personalized PCA (TP-PCA), adapted from the seminal work by \cite{shi2024personalized}. The theoretical analysis reveals that the proposed TPS-PCA estimators reach the minimax lower bound in terms of view-wise tensor decoupling, whereas the TP-PCA estimators only achieve a rate of average decoupling error across views, which is still slower than that of the TPS-PCA estimators. Extensive numerical experiments are conducted on synthetic and real datasets, demonstrating the wide applicability of the proposed method in fields including power management, financial analysis, and activity recognition.

翻译：暂无翻译