  Single-cell data integration can provide a comprehensive molecular view of
cells, and many algorithms have been developed to remove unwanted technical or
biological variations and integrate heterogeneous single-cell datasets. Despite
their wide usage, existing methods suffer from several fundamental limitations.
In particular, we lack a rigorous statistical test for whether two
high-dimensional single-cell datasets are alignable (and therefore should even
be aligned). Moreover, popular methods can substantially distort the data
during alignment, making the aligned data and downstream analysis difficult to
interpret. To overcome these limitations, we present a spectral manifold
alignment and inference (SMAI) framework, which enables principled and
interpretable alignability testing and structure-preserving integration of
single-cell data with the same type of features. SMAI provides a statistical
test to robustly assess the alignability between datasets to avoid misleading
inference, and is justified by high-dimensional statistical theory. On a
diverse range of real and simulated benchmark datasets, it outperforms commonly
used alignment methods. Moreover, we show that SMAI improves various downstream
analyses such as identification of differentially expressed genes and
imputation of single-cell spatial transcriptomics, providing further biological
insights. SMAI's interpretability also enables quantification and a deeper
understanding of the sources of technical confounders in single-cell data.
