Limit Points of Reflow with Minibatch Optimal Transport

What the math gives to ML

The paper gives a constructive interpretation of minibatch OT reflow as an alternating projection-like procedure on couplings: reflow replaces a coupling by the deterministic flow induced by its conditional mean velocity, while minibatch OT replaces each sampled pairing by an optimal assignment within batches. The squared transport cost is monotone non-increasing across both operations, and every limit point is N-cyclically monotone, even when global optimal transport is not solved. This supplies a principled finite-batch pairing strategy for flow-matching training that should reduce trajectory crossings and inference discretization error without requiring a full-dataset OT solve. The most direct ML transfer is to insert repeated minibatch OT reassignment into flow-matching training and use the paper's cost and cyclic-monotonicity quantities as stopping and diagnostic criteria.