  A common theme in causal inference is learning causal relationships between
observed variables, also known as causal discovery. This is usually a daunting
task, given the large number of candidate causal graphs and the combinatorial
nature of the search space. Perhaps for this reason, most research has so far
focused on relatively small causal graphs, with up to hundreds of nodes.
However, recent advances in fields like biology enable generating experimental
data sets with thousands of interventions followed by rich profiling of
thousands of variables, raising the opportunity and urgent need for large
causal graph models. Here, we introduce the notion of factor directed acyclic
graphs (f-DAGs) as a way to restrict the search space to non-linear low-rank
causal interaction models. Combining this novel structural assumption with
recent advances that bridge the gap between causal discovery and continuous
optimization, we achieve causal discovery on thousands of variables.
Additionally, as a model for the impact of statistical noise on this estimation
procedure, we study a model of edge perturbations of the f-DAG skeleton based
on random graphs and quantify the effect of such perturbations on the f-DAG
rank. This theoretical analysis suggests that the set of candidate f-DAGs is
much smaller than the whole DAG space and thus may be more suitable as a search
space in the high-dimensional regime where the underlying skeleton is hard to
assess. We propose Differentiable Causal Discovery of Factor Graphs (DCD-FG), a
scalable implementation of -DAG constrained causal discovery for
high-dimensional interventional data. DCD-FG uses a Gaussian non-linear
low-rank structural equation model and shows significant improvements compared
to state-of-the-art methods in both simulations as well as a recent large-scale
single-cell RNA sequencing data set with hundreds of genetic interventions.
