Probabilistic Count Matrix Factorization for Single Cell Expression Data Analysis

Abstract

The development of high throughput single-cell technologies now allows the investigation of the genome-wide diversity of transcription. This diversity has shown two faces: the expression dynamics (gene to gene variability) can be quantified more accurately, thanks to the measurement of lowly-expressed genes. Second, the cell-to-cell variability is high, with a low proportion of cells expressing the same gene at the same time/level. Those emerging patterns appear to be very challenging from the statistical point of view, especially to represent and to provide a summarized view of single-cell expression data. PCA is one of the most powerful framework to provide a suitable representation of high dimensional datasets, by searching for new axis catching the most variability in the data. Unfortunately, classical PCA is based on Euclidean distances and projections that work poorly in presence of over-dispersed counts that show zero-inflation. We propose a probabilistic Count Matrix Factorization (pCMF) approach for single-cell expression data analysis, that relies on a sparse Gamma-Poisson factor model. This hierarchical model is inferred using a variational EM algorithm. We show how this probabilistic framework induces a geometry that is suitable for single-cell data, and produces a compression of the data that is very powerful for clustering purposes. Our method is competed to other standard representation methods like t-SNE, and we illustrate its performance for the representation of single-cell data. We especially focus on a publicly available data set, being single-cell expression profile of neural stem cells.