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.

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