  Modeling true world data-generating processes lies at the heart of empirical
science. Structural Causal Models (SCMs) and their associated Directed Acyclic
Graphs (DAGs) provide an increasingly popular answer to such problems by
defining the causal generative process that transforms random noise into
observations. However, learning them from observational data poses an ill-posed
and NP-hard inverse problem in general. In this work, we propose a new and
equivalent formalism that does not require DAGs to describe them, viewed as
fixed-point problems on the causally ordered variables, and we show three
important cases where they can be uniquely recovered given the topological
ordering (TO). To the best of our knowledge, we obtain the weakest conditions
for their recovery when TO is known. Based on this, we design a two-stage
causal generative model that first infers the causal order from observations in
a zero-shot manner, thus by-passing the search, and then learns the generative
fixed-point SCM on the ordered variables. To infer TOs from observations, we
propose to amortize the learning of TOs on generated datasets by sequentially
predicting the leaves of graphs seen during training. To learn fixed-point
SCMs, we design a transformer-based architecture that exploits a new attention
mechanism enabling the modeling of causal structures, and show that this
parameterization is consistent with our formalism. Finally, we conduct an
extensive evaluation of each method individually, and show that when combined,
our model outperforms various baselines on generated out-of-distribution
problems.
