Computational and Statistical Guarantees of the \textit{c}-Rectified flow

What the math gives to ML

The paper's central transferable object is a cost-aware rectified-flow iteration that projects a learned velocity field onto a gradient class while retaining prescribed source and target marginals. This supplies a principled alternative to unconstrained flow matching: gradient projection should remove rotational components that waste transport cost, and the paper claims convergence toward the optimal-transport coupling even when source and target covariances do not commute. The most practical neural-network adaptation is a potential-based flow-matching module, optionally iterated with re-coupling, and evaluated against ordinary rectified flow on transport cost, endpoint error, and generation quality.