Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices

Estimating singular subspaces from noisy matrices is a fundamental problem with wide-ranging applications across various fields. Driven by the challenges of data integration and multi-view analysis, this study focuses on estimating shared singular subspaces across multiple matrices within a low-rank matrix denoising framework. A common approach for this task is to perform singular value decomposition on the stacked matrix (Stack-SVD), which concatenates all the matrices. We establish that Stack-SVD achieves minimax rate-optimality when the true singular subspaces of the noisy matrices are identical, whereas a popular alternative approach based on SVD of concatenated singular vector matrices (Average-SVD) can be sub-optimal. We then tackle the more complex scenario where the true singular subspaces are only partially shared across matrices. For various cases of partial sharing, we rigorously characterize the conditions under which Stack-SVD remains effective, achieves minimax optimality, or fails to deliver consistent estimates, offering theoretical insights into its practical applicability. To address the limitations of Stack-SVD in scenarios with partial sharing, we propose novel estimators and an efficient algorithm designed to identify both shared and unshared singular vectors. We further prove that these methods attain minimax rate-optimality under partial sharing. Extensive simulations and real-world data applications demonstrate the advantages of our proposed approach.