Graph Quotient Minibatch Coupling

Integrate this into the minibatch sampler immediately before flow-matching target construction. For each batch, compute a B-by-B cost matrix. For every pair (a,b), obtain an approximate node permutation with a cheap graph matcher; for an MVP, initialize node correspondences from degree and node-feature similarity, run 5-20 Sinkhorn iterations on an edge-consistency objective, and round the resulting soft matrix to a permutation with the Hungarian algorithm. Set C[a,b] to the aligned squared edge-tensor distance. Compute the outer coupling P with Sinkhorn iterations using uniform source and target marginals and entropy epsilon. Stop gradients through P and the discrete permutations initially. Either train on all B-squared pairs with weights B*P[a,b], or sample pairs from the normalized P distribution to preserve model-evaluation cost. For each selected pair, permute Y_b, sample t, construct Z=(1-t)*X_a+t*Y_aligned, and regress to Y_aligned-X_a. The paper-derived quantities are quotient pair costs and the doubly stochastic coupling; matching quality and epsilon are empirical approximations. Begin with B=16 or 32, N<=32, a small equivariant graph flow model, and a molecular dataset. Compare random pairings, independently nearest-neighbor pairings, and quotient coupling at equal compute. Measure displacement, gradient cosine similarity, validation loss, and generated validity under 4/8/16-step Euler sampling. Success means lower displacement and gradient conflict with improved sample quality; if O(B^2) matching dominates, retain only top-k candidates per source.