  Important tasks in the study of genomic data include the identification of
groups of similar cells (for example by clustering), and visualisation of data
summaries (for example by dimensional reduction). In this paper, we propose a
novel approach to studying single-cell genomic data, by modelling the observed
genomic data count matrix $\mathbf{X}\in\mathbb{Z}_{\geq0}^{p\times n}$ as a
bipartite network with multi-edges. Utilising this first-principles network
representation of the raw data, we propose clustering single cells in a
suitably identified $d$-dimensional Laplacian Eigenspace (LE) via a Gaussian
mixture model (GMM-LE), and employing UMAP to non-linearly project the LE to
two dimensions for visualisation (UMAP-LE). This LE representation of the data
estimates transformed latent positions (of genes and cells), under a latent
position model of nodes in a bipartite stochastic network. We demonstrate how
these estimated latent positions can enable fine-grained clustering and
visualisation of single-cell genomic data, by application to data from three
recent genomics studies in different biological contexts. In each data
application, clusters of cells independently learned by our proposed
methodology are found to correspond to cells expressing specific marker genes
that were independently defined by domain experts. In this validation setting,
our proposed clustering methodology outperforms the industry-standard for these
data. Furthermore, we validate components of the LE decomposition of the data
by contrasting healthy cells from normal and at-risk groups in a
machine-learning model, thereby identifying an LE cancer biomarker that
significantly predicts long-term patient survival outcome in two independent
validation cohorts with data from 1904 and 1091 individuals.
