  The need for multimodal data integration arises naturally when multiple
complementary sets of features are measured on the same sample. Under a
dependent multifactor model, we develop a fully data-driven orchestrated
approximate message passing algorithm for integrating information across these
feature sets to achieve statistically optimal signal recovery. In practice,
these reference data sets are often queried later by new subjects that are only
partially observed. Leveraging on asymptotic normality of estimates generated
by our data integration method, we further develop an asymptotically valid
prediction set for the latent representation of any such query subject. We
demonstrate the prowess of both the data integration and the prediction set
construction algorithms on a tri-modal single-cell dataset.
