Many causal inference problems require sampling from interventional outcome distributions rather than estimating low-dimensional summaries. Existing approaches often rely on fitting generative models of the target distribution together with additional nuisance components, which can be computationally demanding for high-dimensional outcomes. We propose Interventional Maximum Mean Discrepancy Flow (IMMD-Flow), a particle-based method that instead represents the target interventional distribution through its kernel mean embedding. This kernel embedding admits a doubly robust, Neyman-orthogonal estimator with finite-sample guarantees and can be efficiently reused to sample from the interventional distribution via MMD gradient flow. IMMD-Flow accommodates both hard and soft interventions, admits theoretical guarantees for estimation and particle approximation, and extends to high-dimensional image outcomes through frozen encoders. Across image-based benchmarks, IMMD-Flow matches or improves upon state-of-the-art generative approaches while showing significant speed advantages.
Robin Leman, Bariscan Bozkurt, Sofiane Tanji, Arthur Gretton, Houssam Zenati
We propose a test of whether two interventions induce the same outcome distribution when outcomes are structured and high-dimensional, such as images, text and graphs. We represent the interventional distributions as mean embeddings on learned deep features, such that the treatment effect is a doubly robust representation of the difference in mean embeddings. Our test, DR-KTE-Deep, optimises the learned features so as to maximise an estimated power criterion on this treatment effect representation. To motivate our learning objective, we show that the power criterion characterises the asymptotic signal-to-noise ratio of the test under fixed alternatives, and we derive a uniform finite-sample bound for its regularised estimate. We show that, with the learned kernel fixed, a held-out test is asymptotically valid under the null given suitable nuisance-estimation rates. Empirically, DR-KTE-Deep achieves substantially higher power than fixed and bandwidth-optimised kernel baselines across image, text and graph outcomes, while maintaining well-calibrated Type-I error.