Multi-source data integration problems over datasets from different sources covering different but possibly overlapping sets of entities have become increasingly important in many real-world areas, including genomics, single-cell analysis, and healthcare research. 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 from different sources are only identifiable up to some transformation, how to align them across sources using the sources' overlapping entities becomes a key challenge. Existing methods align the sources in a sequential or tree-structured manner, and are therefore sensitive to the chosen order and exploit only part of the available overlapping information. Motivated by this limitation, 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, thereby making full use of all overlapping information across the sources. We develop an efficient iterative algorithm for GSMMI that is fast and scalable to the large-scale data arising in these applications. We show both theoretically and empirically that GSMMI improves alignment accuracy, with clear improvements even under modest overlap structure. Moreover, we develop GSMMI to be broadly applicable across data types, covering symmetric positive semidefinite, symmetric indefinite, and asymmetric or rectangular matrices, and even settings where sources overlap only in their rows or only in their columns, making it suitable for a wide variety of application scenarios.

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