MOTIVATION: Networks underlie the generation and interpretation of many biological datasets: gene networks shed light on the regulatory structure of the genome, and cell networks can capture structure of the tumor micro-environment. However, most methods that learn such networks make the faulty "independence assumption"; to learn the gene network, they assume that no cell network exists. "Multi-axis" methods, which do not make this assumption, fail to scale beyond a few thousand cells or genes. This limits their applicability to only the smallest datasets. RESULTS: We develop a multi-axis method, which learns conditional dependency networks, capable of processing million-cell datasets within minutes. This was previously impossible, and unlocks the use of such methods on modern scRNA-seq datasets, as well as more complex datasets. We apply the method to a new scRNA-seq dataset for neuronal cell development, and compare the result to an existing state of the art method, hdWGCNA. We demonstrate that the new method yields gene networks that have a more focused biological interpretation and that the simultaneously learned cell network has advantages over a conventional kNN-based clustering. Further, our method yields novel biological insights by identifying long non-coding RNAs that potentially have a role in neuronal development. AVAILABILITY AND IMPLEMENTATION: Our methodology is available as a Python package GmGM on PyPI (https://pypi.org/project/GmGM/0.5.3/). The code for all experiments performed in this article is available on GitHub (https://github.com/BaileyAndrew/GmGM-Bioinformatics) and Zenodo (10.5281/zenodo.20384566).

The independence assumption between random variables is a useful tool to increase the tractability of a modelling framework. However, this assumption can be too simplistic; failing to take dependencies into account can cause models to fail dramatically. The field of multi-axis graphical modelling (also called multi-way modelling, Kronecker-separable modelling) has seen growth over the past decade, but these models require that the data have zero mean. In the multi-axis case, inference is typically done in the single sample scenario, making mean inference impossible. In this paper, we demonstrate how the zero-mean assumption can cause egregious modelling errors for Kronecker-sum-decomposable Gaussian graphical models, as well as propose a relaxation to the zero-mean assumption that allows the avoidance of such errors. Specifically, we propose the "Kronecker-sum-structured mean" assumption, which leads to models with nonconvex-but-unimodal log-likelihoods that can be solved efficiently with coordinate descent.

Multi-axis graphical modelling techniques allow us to perform network inference without making independence assumptions. This is done by replacing the independence assumption with a weaker assumption about the interaction between the axes; there are several choices for which assumption to use. In single-cell RNA sequencing data, genes may interact differently depending on whether they are expressed in the same cell, or in different cells. Unfortunately, current methods are not able to make this distinction. In this paper, we address this problem by introducing the strong product model for Gaussian graphical modelling.