Multi-source data integration problems over datasets from different sources covering different but possibly overlapping sets of entities have become increasingly important in many applications, including genomics, single-cell analysis, and healthcare research. We study a model of multi-source data integration where we observe overlapping submatrices of a low-rank matrix, each corrupted by additive noise. In such problems, one often first learns a low-dimensional representation of the entities within each source and then integrates these representations across sources. As the representations are only identifiable up to some transformation, aligning them using the sources' overlapping entities becomes a key challenge. Existing methods align these estimates in a sequential or tree-structured manner, and are therefore sensitive to the chosen order and exploit only part of the overlapping information. We propose Global Synchronized Multiple Matrix Integration (GSMMI), which formulates this alignment problem as a global synchronization problem and jointly aligns all sources using all pairwise overlaps at once. We develop an efficient, scalable iterative algorithm for GSMMI, and show both theoretically and empirically that it improves alignment accuracy, with clear gains over prior approaches even under moderate overlap. We additionally establish information-theoretic and computational lower bounds for detection in data integration, and find GSMMI to be empirically near-optimal. GSMMI applies broadly across symmetric positive semidefinite, symmetric indefinite, and asymmetric or rectangular matrices, as well as settings where sources overlap only in rows or only in columns.

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