Causal abstraction offers a principled framework for mechanistic interpretability, aligning a high-level causal model with the low-level computation realized by a neural network through counterfactual intervention analysis. Existing methods such as distributed alignment search (DAS) learn expressive subspace interventions, but the relevant neural site is unknown a priori, so finding a handle requires a computationally burdensome search over candidate sites. We introduce PLOT (Progressive Localization via Optimal Transport), a transport-based framework that localizes causal variables from the output effect geometry of abstract and neural interventions. PLOT fits an optimal transport coupling between abstract variables and candidate neural sites, yielding a global soft correspondence that can be calibrated into intervention handles. In simple settings, a single coupling over individual neurons suffices. In larger models, PLOT is applied progressively, moving from coarse sites such as tokens, timesteps, or layers to finer supports such as coordinate groups or PCA spans, and optionally guiding DAS based on the localized signal. Across experiments of increasing complexity, transport-only PLOT handles are exceedingly fast and competitive on accuracy, while PLOT-guided DAS reaches DAS-level accuracy at a fraction of full DAS runtime, providing an efficient localization engine for causal abstraction research at scale.

These appendices contain demonstrations of the results in the main text as well as additional technical notes.

The variables of the low-level model (left) are divided into partitions (center) such that each low-level partition corresponds to a high level variable from the high-level model (right). The circles represent variables and the arrows represent causal dependencies. Blue circles are variables that are not being intervened on and red circles are variables that are being intervened on. Observe that a low-level causal dependence between partitions does not necessarily result in a high-level causal dependence between variables and that not every low-level intervention results in a high level intervention.

We have shown experimentally that our method is effective in a variety of domains; however, other problem domains may require additional hyperparameter tuning, which can be expensive.