Causal Abstraction of Linear Systems: Sufficient Conditions and Methods

Causal systems often admit multiple levels of description, where micro-level variables are aggregated into macro-level representations. A valid causal abstraction (CA) must preserve interventional semantics while producing a more compact, interpretable model. We study the problem of learning CAs for linear structural causal models (SCMs), formulating it as a two-stage pipeline: graph partitioning followed by linear aggregation weight learning. For the first stage, we establish sufficient graphical conditions for exact CA and propose three partitioning algorithms including one that guarantees these conditions. For the second stage, we extend algebraic characterizations of CA to a subclass of cyclic SCMs and introduce a differentiable objective for learning aggregation weights as a constrained optimization problem. Together, our end-to-end framework reduces a $p$-variable system to a target dimension $k<p$. Experiments on synthetic graphs and empirical economic input-output networks demonstrate that our approach is able to recover non-trivial abstractions and outperforms existing unsupervised baselines while offering a principled trade-off between exactness and informativeness.