🤖 AI Summary
This work addresses the numerical instability and spurious correlations that arise in deterministic generative models under causal interventions, particularly when traversing low-density regions of the data distribution. To mitigate support mismatch induced by out-of-distribution interventions, we propose a Causal Schrödinger Bridge framework that formulates counterfactual reasoning as a structurally constrained entropy-regularized optimal transport problem. By constructing robust diffusion paths via stochastic differential equations, our approach ensures stable counterfactual generation. We establish a structural decomposition theorem that factorizes high-dimensional counterfactual bridging into locally robust transitions while rigorously preserving structural admissibility constraints. Experiments on high-dimensional interventions in Morpho-MNIST demonstrate that our method significantly outperforms deterministic baselines, maintaining superior structural consistency even under strong out-of-distribution shifts.
📝 Abstract
Generative modeling typically seeks the path of least action via deterministic flows (ODE). While effective for in-distribution tasks, we argue that these deterministic paths become brittle under causal interventions, which often require transporting probability mass across low-density regions ("off-manifold") where the vector field is ill-defined. This leads to numerical instability and spurious correlations. In this work, we introduce the Causal Schr\"odinger Bridge (CSB), a framework that reformulates counterfactual inference as Entropic Optimal Transport. Unlike deterministic approaches that require strict invertibility, CSB leverages diffusion processes (SDEs) to robustly"tunnel"through support mismatches while strictly enforcing structural admissibility constraints. We prove the Structural Decomposition Theorem, showing that the global high-dimensional bridge factorizes into local, robust transitions. Empirical validation on high-dimensional interventions (Morpho-MNIST) demonstrates that CSB significantly outperforms deterministic baselines in structural consistency, particularly in regimes of strong, out-of-distribution treatments.