Central limit theorem for Wasserstein projection - the case of convex order

What the math gives to ML

The paper provides a constructive statistical geometry for projecting a distribution onto the cone of measures below a reference distribution in convex order, using an optimal-transport cost. The transferable asset is that convex order simultaneously controls expectations of every convex test function, so one projection can replace manually chosen variance, tail, and risk penalties. A practical neural-network use is a differentiable distributional-output layer that projects student predictions onto the convex-order cone of teacher predictions or calibrated target distributions. This is most plausible for distributional regression and uncertainty-aware distillation, where controlling spread without selecting many separate convex penalties may improve calibration and robustness.