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

Abstract

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.

Discussion of “Exponential-Family Embedding With Application to Cell Developmental Trajectories for Single-Cell RNA-seq Data”

“…To see relevance of this variability, we also applied SVD (with k = 5 latent dimension) and UMAP (using the umap R package with k = 2 latent dimensions) followed by the same k-means clustering procedure (5 clusters) to the same data. Consistent with Lin, Lei, and Roeder (2021), SVD and UMAP were run using normalized and log 2 -transformed read counts, whereas eSVD was run using normalized data without log 2 -transformation. It was observed that the variability in eSVD's performance due to gene filtering can be even bigger than the variability across different dimension reduction methods (Figure 1(c)).…”

“…Thanks to the rapidly evolving technology, the data size also grows fast. Lin, Lei, and Roeder (2021) has demonstrated eSVD using data with n = 10 2 − 10 3 cells and p = 10 2 − 10 3 genes. However, today's cell atlas projects increasingly generate much larger datasets, such as scRNA-seq data consisting of n = 10 5 − 10 6 cells and p = 10 3 − 10 4 genes (Regev et al 2017;Su et al 2020) or single-cell chromatin accessibility data consisting of n = 10 4 − 10 5 cells and p = 10 4 − 10 6 genomic regulatory elements (Cusanovich et al 2018).…”

“…We would like to congratulate Lin, Lei, and Roeder on developing an elegant framework, exponential-family singular value decomposition (eSVD) (Lin, Lei, and Roeder 2021), for embedding multivariate data into a lower dimensional space. Lowdimensional embedding is an important dimension reduction tool with many applications in high-dimensional data analyses.…”

“…Examples include data compression, visualization, denoising, feature extraction, improving computational efficiency, and mitigating the curse of dimensionality for downstream data analysis tasks. While many low-dimensional embedding methods have been developed in the past, the work by Lin, Lei, and Roeder (2021) introduces an appealing new solution for at least two reasons.…”

“…By providing a flexible low-dimensional embedding framework amenable to theoretical analysis, eSVD adds a very valuable dimension reduction tool to the toolbox of data analysts. Clearly, the work by Lin, Lei, and Roeder (2021) only represents a start. To unleash the full potential of eSVD, there are still a number of open problems and challenges that need to be addressed in the future.…”