Source
Zeng, Du, Zhang, Xia, Liu and Tong (2024), LIT-1814, Definition 2 (pp. 2–3), Proposition 1 and Theorem 1 (pp. 3, 5; proofs pp. 11–12 of arXiv 2310.04470v2); read in NOTE-1334.
What was actually shown
The identity. For a coupling tensor S with marginals µ₁, …, µ_K and q = 2, both terms of the MFGW objective depend on S only through its two-way marginals P_{j,k}(S): the Wasserstein term because the cost tensor is a sum of pairwise costs (eq. 14, p. 12), the GW term because each pair’s distortion integrates out the other coordinates (eq. 15). So the objective is Σ_{j<k} FGW(G_j, G_k; P_{j,k}(S)) (p. 12). Each marginal is a feasible pairwise coupling, hence Theorem 1: Σ_{j<k} FGW(G_j, G_k) ≤ MFGW(G₁, …, G_K) (p. 5).
Checked. For K = 3 random 4-node structures with random embeddings, the objective at a random feasible tensor equalled the sum of pairwise objectives at its marginals to machine precision, and the tensor form of Proposition 1 matched the direct sum over j < k (numpy; my script). The sums must run over j < k: Definition 2 prints 1 ≤ j, k ≤ K, which would double both terms.
The equality condition (the record’s). MFGW = min over S of Σ F_{jk}(P_{jk}(S)) and the right side is Σ min F_{jk}, so they are equal iff some tensor has every two-way marginal pairwise optimal. Pairwise couplings fitted independently need not be the marginals of any tensor; the gap is a measure of that failure.
Composition (the record’s). If S is supported on tuples given by bijections, its marginals are permutations and P₁₃ is P₂₃ after P₁₂. For soft couplings the marginals are jointly realizable but need not compose. With two nodes per network and S = ¼ on (1,1,1), (1,2,1), (2,1,2), (2,2,2), P₁₂ and P₂₃ are uniform, so their composition is uniform, while P₁₃ is the identity.
What could have failed. The joint objective could have contained genuinely K-way terms, coupling three networks at once. Under square loss with a pairwise-sum cost it does not; all its content is pairwise plus the constraint of joint realizability.
What this does not say
- That joint alignment is more accurate. That is the empirical claim of THEORY-1159, and the inequality is silent on it.
- That consistency is free. Joint realizability is bought by a higher objective whenever the pairwise optima conflict; whether the conflict is noise or a true non-transitivity is not settled by the mathematics.
- That HOT’s output composes. Its tensors are entropic and soft (eqs. 11–13), so composition holds only in the deterministic limit.
- That the gap was measured. The paper reports no value of MFGW − Σ FGW on any data.
- Anything for a K-way cost. With a cost tensor that is not a sum of pairwise terms, the identity does not hold.