Exponential-family embedding with application to cell developmental trajectories for single-cell RNA-seq data

Scientists often embed cells into a lower-dimensional space when studying single-cell RNA-seq data for improved downstream analyses such as developmental trajectory analyses, but the statistical properties of such non-linear embedding methods are often not well understood. In this article, we develop the eSVD (exponential-family SVD), a non-linear embedding method for both cells and genes jointly with respect to a random dot product model using exponential-family distributions. Our estimator uses alternating minimization, which enables us to have a computationally-efficient method, prove the identifiability conditions and consistency of our method, and provide statistically-principled procedures to tune our method. All these qualities help advance the single-cell embedding literature, and we provide extensive simulations to demonstrate that the eSVD is competitive compared to other embedding methods. We apply the eSVD via Gaussian distributions where the standard deviations are proportional to the means to analyze a single-cell dataset of oligodendrocytes in mouse brains (Marques et al., 2016). Using the eSVD estimated embedding, we then investigate the cell developmental trajectories of the oligodendrocytes. While previous results are not able to distinguish the trajectories among the mature oligodendrocyte cell types, our diagnostics and results demonstrate there are two major developmental trajectories that diverge at mature oligodendrocytes.