Source
Yu, Qiu, Zeng, Thai, Liu and Tong (2026), LIT-1807, §4.1 and Appendix A of arXiv 2605.24395v2; read in NOTE-1327.
What was actually shown
Derivation (A.1, pp. 14–16). The optimal entropic plan is T_ij = exp((α_i + β_j − C̃_ij)/ε) (eq. 13). Differentiating gives dT = (1/ε) T ⊙ (dα1ᵀ + 1dβᵀ − dC̃) (eq. 15), and keeping the marginals fixed gives A(dα, dβ) = ((T ⊙ dC̃)1, (T ⊙ dC̃)ᵀ1) (eq. 24). An adjoint vector y solving Ay = ((T ⊙ ∇_T f)1, (T ⊙ ∇_T f)ᵀ1) eliminates dα, dβ, so ∇_C̃ f = (1/ε) T ⊙ (y_α1ᵀ + 1y_βᵀ − ∇_T f) (eq. 32).
Solvability (A.2–A.3, pp. 16–17). zᵀAz = Σ T_ij(s_i + t_j)² ≥ 0; (1, −1) is in the null space; the right side is orthogonal to it, so conjugate gradients converge (Kaasschieter 1988, Hayami 2018). Any two solutions differ along null directions, which cancel in y_α1ᵀ + 1y_βᵀ on the support of T, so the gradient is unique. For a fully supported (dense entropic) plan the null space is exactly span(1, −1); for a plan with disconnected support it is larger, and the gradient is still unique on the support.
Use. Supervising pair (i, j) moves C̃_ij by −βC_ij, so its first-order effect on f is −βC_ij(∇_C̃ f)_ij (eq. 7, p. 5).
Checked. On a 6 × 5 problem with ε = 0.1, solved by log-domain Sinkhorn to convergence, the formula matched central finite differences to 9 × 10⁻¹¹ (f = ‖T‖²) and 6 × 10⁻¹⁰ (f = Σ T log T), and A had rank n + m − 1 (numpy/scipy; my script).
What could have failed. The singular system could have had a right side outside the range of A, or solutions giving different gradients. Neither happens.
What this does not say
- What a label actually does. A label is a finite change, a cost set to (1 − β)C_ij, β = 1 by default; the derivative predicts it only locally.
- Anything for unregularized OT, whose plan is piecewise constant in the cost; the 1/ε factor shows the derivative grows as ε → 0.
- Anything exact for aligners whose cost depends on the plan, such as fused Gromov–Wasserstein, or whose plan is not the fixed-cost entropic solution. The paper applies the score to such aligners without saying how (NOTE-1327).
- That the utility measures accuracy. The formula differentiates whatever f is supplied.
- That this is new mathematics. Implicit differentiation of entropic OT is known; the contribution is its use to score queries.