Cycle-Monotonicity Regularizer for Pairing and Velocity Training

Attach the regularizer to the pairing logits or to predicted endpoints in the flow-matching batch. Start with a batch of N source-target pairs (x_i,y_i), where x_i are latent samples and y_i are data samples; if the assignment is learned, let P[i,j] be Sinkhorn-normalized matching probabilities and use soft expected target embeddings ilde y_i=\sum_jP[i,j]y_j. Sample 16 to 64 random cycles per batch, with k uniformly chosen from 2 through min(N,6), because enumerating all cycles is unnecessary. Compute Delta_k exactly from the displayed inner products, then accumulate relu(m-Delta_k)^2. Use m=0 initially and lambda_cycle in {1e-3,1e-2,1e-1}; normalize Delta by the average squared endpoint norm so the coefficient is dimensionless. Add this term to the standard flow-matching regression loss, backpropagating through P if using soft assignments, or through a learned endpoint predictor if the pairing is fixed. The paper provides the target inequality and its connection to minibatch OT limits; the engineer must estimate violation rates and choose cycle samples empirically. For a hard-assignment implementation, use the loss only as a diagnostic or apply it to a differentiable Sinkhorn relaxation. First test on two-dimensional Gaussian mixtures and a CIFAR-10 2D latent projection with a small MLP, comparing standard flow matching against minibatch OT and minibatch OT plus the penalty. Track transport cost, fraction of negative Delta_k values, velocity-field regression loss, trajectory curvature, and FID or sliced Wasserstein distance. A useful success criterion is at least a 30% reduction in cycle violations and lower one-step or four-step sampling error without increasing training FLOPs by more than 10%.