  The development of high throughput single-cell sequencing technologies now
allows the investigation of the population level diversity of cellular
transcriptomes. This diversity has shown two faces. First, 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 latent directions 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 drop-out events
(zero-inflation) like single-cell expression data 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. It is able to jointly build
a low dimensional representation of cells and genes. We show how this
probabilistic framework induces a geometry that is suitable for single-cell
data visualization, and produces a compression of the data that is very
powerful for clustering purposes. Our method is competed against other standard
representation methods like t-SNE, and we illustrate its performance for the
representation of single-cell data. We especially focus on publicly available
single-cell RNA-seq datasets.
