--- title: Excess Wasserstein Gap in Optimal Transport url: https://www.emergentmind.com/topics/excess-wasserstein-gap type: topic --- # Excess Wasserstein Gap in Optimal Transport “Excess Wasserstein gap” is best understood as an *Editor’s term* for a family of residual quantities that measure how far a tractable object remains from an exact Wasserstein quantity. In the literature considered here, the expression is not introduced uniformly, but closely related discrepancies appear in several precise forms: the residual between a continuous-time Wasserstein gradient flow and its variational discretization, the difference between a surrogate Gaussian transport cost and the exact Gaussian Wasserstein cost, the excess of a distributional distance over a mean-gap summary, the deficit between full and sliced Wasserstein costs, and the mismatch between a conditional Wasserstein objective and its implementable adversarial formulation [2310.20090] [2603.28992] [2508.17235] [2605.25453] [2103.13906]. ## 1. Scope of the notion Across these works, the relevant quantity is always a discrepancy between a more exact transport object and a reduced representation. The sign convention varies by context, but the structural role is stable: the gap isolates what is lost under discretization, projection, surrogation, or aggregation. | Context | Precise quantity | Interpretation | |---|---|---| | VI and gradient flows [2310.20090] | continuous-time Wasserstein gradient flow vs. discrete-time BBVI update | discretization and parameterization residual | | Gaussian mixture flow matching [2603.28992] | \(C_{ij}-W_{2,ij}^2\) | excess of surrogate transport cost over exact Gaussian Wasserstein cost | | Life tables [2508.17235] | \(W_1(P_A,P_B)-|e_{0,A}-e_{0,B}|\) | distributional excess beyond life expectancy gap | | Sliced Wasserstein [2605.25453] | \(\mathrm D(\mu,\nu)=\frac1dW_2^2(\mu,\nu)-SW_2^2(\mu,\nu)\) | multidimensional transport cost not captured by slicing | | Conditional WGANs [2103.13906] | averaged conditional Wasserstein objective vs. practical discriminator objective | objective mismatch if \(\mathbb E\) and \(\sup\) are not exchangeable | This suggests that the phrase is not a single invariant of optimal transport, but rather a context-dependent descriptor for the residual left after replacing a full Wasserstein object by something computationally or analytically simpler. ## 2. Variational inference and Wasserstein gradient-flow residuals In “Bridging the Gap Between Variational Inference and Wasserstein Gradient Flows” [2310.20090], the relevant gap is the mismatch between the continuous-time Wasserstein gradient flow and the discrete-time black-box variational inference update. The paper studies a Bures-Wasserstein gradient flow on a Gaussian variational family, \[ q_\theta(z)=\mathcal N(z;\mu,\Sigma), \] and shows that, under suitable conditions, the Bures-Wasserstein gradient flow can be recast as a Euclidean gradient flow whose forward Euler discretization is exactly the standard black-box variational inference update. The forward Euler step is written as \[ \theta_{k+1}=\theta_k-\eta\,\nabla_\theta \mathcal L(\theta_k), \] or, equivalently for ELBO maximization, \[ \theta_{k+1}=\theta_k+\eta\,\widehat{\nabla_\theta \mathrm{ELBO}(\theta_k)}. \] The paper’s key claim is that this discrete-time VI update can be interpreted as a forward Euler step for the Wasserstein/Bures-Wasserstein flow. In that sense, the excess gap is not a new divergence but the residual between ideal continuous transport dynamics and the algorithmic update used in practice. A second key object is the path-derivative gradient estimator. In reparameterized form, \[ z=g_\theta(\varepsilon), \qquad \varepsilon\sim p(\varepsilon), \] and \[ \nabla_\theta \mathbb E_{\varepsilon}\big[f(g_\theta(\varepsilon))\big] = \mathbb E_{\varepsilon}\big[\nabla_z f(g_\theta(\varepsilon))\,\nabla_\theta g_\theta(\varepsilon)\big]. \] The paper identifies the vector field of the gradient flow with this path-derivative gradient estimator. It also frames the path-derivative gradient as a distillation procedure: the Wasserstein gradient flow is the “teacher” dynamics in probability space, while the variational family acts as a “student” that mimics that flow. Within this framework, a natural excess Wasserstein gap is the deviation between the algorithmic update and the exact flow vector field, written in the synthesis as \[ \text{gap} \approx \left\| \frac{\theta_{k+1}-\theta_k}{\eta} + \nabla^{\mathrm{BW}}\mathcal L(\theta_k) \right\|. \] The paper’s equivalence is strongest in the Gaussian case and relies on a Gaussian variational family, Bures-Wasserstein geometry, sufficient regularity of the target density and variational objective, a reparameterizable family, and the Euclidean recasting that is special to the Gaussian/Bures setting. The same distillation viewpoint is then extended to \(f\)-divergences and non-Gaussian variational families, yielding a new gradient estimator for \(f\)-divergences that is readily implementable in PyTorch or TensorFlow [2310.20090]. ## 3. Surrogate-versus-exact Gaussian transport costs In “An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds” [2603.28992], the excess Wasserstein gap is introduced explicitly as the difference between a surrogate kinetic transport cost and the exact Gaussian Wasserstein cost. The source and target are Gaussian mixture models, \[ \rho_0(x)=\sum_{i=1}^{K_0} a_i \mathcal N(x\mid \mu_i^0,\Sigma_i^0),\qquad \rho_1(x)=\sum_{j=1}^{K_1} b_j \mathcal N(x\mid \mu_j^1,\Sigma_j^1), \] and transport is built component-wise. For each pair \((i,j)\), the paper uses the linear Gaussian path \[ \mu_{ij}(t)=(1-t)\mu_i^0+t\mu_j^1,\qquad \Sigma_{ij}(t)=(1-t)\Sigma_i^0+t\Sigma_j^1. \] A lemma states that if \(\rho_t=\mathcal N(x\mid m(t),\Sigma(t))\) with differentiable \(m(t)\) and \(\Sigma(t)\), then the continuity equation admits the affine solution \[ v(t,x)=\dot m(t)+\frac12\,\dot\Sigma(t)\Sigma(t)^{-1}(x-m(t)). \] Hence the component-wise surrogate field is \[ v_{ij}(t,x)=\dot\mu_{ij}(t)+\frac12\,\dot\Sigma_{ij}(t)\Sigma_{ij}(t)^{-1}(x-\mu_{ij}(t)). \] The surrogate pairwise cost is the kinetic action \[ C_{ij}=\int_0^1 \mathbb E\|v_{ij}(t,X)\|^2\,dt, \qquad X\sim \mathcal N(\mu_{ij}(t),\Sigma_{ij}(t)), \] and the exact comparison target is the Gaussian squared \(2\)-Wasserstein distance \[ W_2^2\Big(\mathcal N(\mu_i^0,\Sigma_i^0),\mathcal N(\mu_j^1,\Sigma_j^1)\Big) = \|\mu_j^1-\mu_i^0\|^2 +\operatorname{Tr}(\Sigma_i^0) +\operatorname{Tr}(\Sigma_j^1) -2\operatorname{Tr}\!\Big(\big((\Sigma_i^0)^{1/2}\Sigma_j^1(\Sigma_i^0)^{1/2}\big)^{1/2}\Big). \] The excess Wasserstein gap is then \[ C_{ij}-W_{2,ij}^2, \] or its absolute value. The analysis is organized by the whitened covariance perturbation size \[ \hat\rho = \left\| (\Sigma_i^0)^{-1/2}\Delta\Sigma_{ij}(\Sigma_i^0)^{-1/2} \right\|, \qquad \Delta\Sigma_{ij}=\Sigma_j^1-\Sigma_i^0. \] Under the local commuting regime \(\hat\rho<1\), the surrogate and exact costs share the same mean term and the same second-order covariance term, \[ \|\Delta\mu_{ij}\|^2+\frac14\operatorname{Tr}\!\big(\Delta\Sigma_{ij}(\Sigma_i^0)^{-1}\Delta\Sigma_{ij}\big), \] so that \[ C_{ij}-W_{2,ij}^2=O(\|C_0\|^3). \] The paper further gives an explicit cubic error bound \[ |C_{ij}-W_{2,ij}^2| \le \bigl(B_C(\Sigma_i^0,\hat\rho)+B_W(\Sigma_i^0,\hat\rho)\bigr)\, \|\Delta\Sigma_{ij}\|^3, \] and emphasizes that the constants deteriorate as \(\Sigma_i^0\) becomes ill-conditioned. For nonlocal regimes, path splitting subdivides the covariance path into local segments. This restores locality of the comparison, but the paper makes an important qualification: path splitting fixes the locality requirement and does not remove the commuting assumption used in the cubic-gap theorem. The exact reference construction, called Method B, uses the Gaussian Wasserstein geodesic and has kinetic action exactly equal to the Gaussian Wasserstein cost. The practical regime map summarized in the paper is that the surrogate is best in local, well-conditioned, and approximately commuting regimes, whereas the exact method is preferable in highly nonlocal, strongly noncommuting, ill-conditioned, or near-boundary regimes [2603.28992]. ## 4. Distributional excess beyond mean-gap summaries In “On the relationship between the Wasserstein distance and differences in life expectancy at birth” [2508.17235], the excess Wasserstein gap is the amount by which a full distributional distance exceeds the absolute mean-gap summary. The paper treats a life table as an age-at-death distribution, with survivorship function \(l(x)\), density \(f(x)\), and distribution function \(F(x)=1-l(x)\). Life expectancy at birth satisfies \[ e_0=\int_0^\omega l(x)\,dx, \] and for two populations \(A\) and \(B\), \[ e_{0,A}-e_{0,B} = \int_0^\omega \bigl(l_A(x)-l_B(x)\bigr)\,dx. \] For one-dimensional age-at-death distributions, the \(1\)-Wasserstein distance simplifies to \[ W_1(P_A,P_B) = \int_0^\omega |F_A(x)-F_B(x)|\,dx = \int_0^\omega |l_A(x)-l_B(x)|\,dx. \] Thus \(W_1\) is the area between the survivorship curves, while the life-expectancy difference is the signed area between them. The main theorem is that if the survivorship functions do not cross, then the \(1\)-Wasserstein distance equals the difference in life expectancy at birth. If, for example, \[ l_A(x)\ge l_B(x)\quad\text{for all }x, \] then \[ W_1(P_A,P_B)=e_{0,A}-e_{0,B}, \] and with reversed ordering \(W_1=|e_{0,A}-e_{0,B}|\). The excess gap arises when curves cross. The synthesis defines it as \[ W_1(P_A,P_B)-|e_{0,A}-e_{0,B}|. \] Because \(W_1\) uses absolute area while \(e_0\) uses signed area, crossing generates cancellation in the mean but not in the transport distance. Therefore, \[ W_1(P_A,P_B)\ge |e_{0,A}-e_{0,B}|, \] with strict inequality whenever survivorship curves cross and cancellation occurs. Demographically, \(e_0\) captures the net mean lifespan difference, whereas \(W_1\) captures the full distributional dissimilarity in ages at death. The empirical sections reinforce this interpretation. For 5,000 sampled country pairs from the Human Mortality Database, the distributions of \(W_1\) and \(e_0\) differences overlap strongly, both range from \(0\) to about \(30.38\) years, the mean \(W_1\) is \(4.18\), the mean \(e_0\) difference is \(3.95\), and Pearson’s \(r=0.99\). A crossing example is England & Wales versus Iceland in 1849: the \(e_0\) values are both about \(37.3\) years, yet \(W_1\) is large because Iceland has higher infant mortality and lower mortality at older ages. For women and men in period life tables from 1990 to 2020, the relationship is even tighter: the average absolute difference between \(W_1\) and \(e_0\) differences is below \(0.5\) in every year, and for women versus men specifically it is below \(0.02\) each year, with maximum difference about \(0.13\) years in 2008 [2508.17235]. ## 5. Sliced Wasserstein deficits as multidimensional excess In “Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit” [2605.25453], the excess Wasserstein gap is formalized as a deficit: \[ \mathrm D(\mu,\nu):=\frac1d\,W_2^2(\mu,\nu)-SW_2^2(\mu,\nu)\ge 0. \] Here \(SW_2\) averages one-dimensional quadratic Wasserstein distances over linear projections. The elementary comparison \[ SW_2^2(\mu,\nu)\le \frac1d W_2^2(\mu,\nu) \] follows by projecting couplings and averaging \[ \int_S |\theta\cdot z|^2\,d\sigma(\theta)=\frac1d|z|^2. \] The deficit is therefore the excess left after accounting for all one-dimensional projections. A central identity shows that, when \(\mu\ll\mathcal L^d\) and \(T=\nabla\varphi\) is the Brenier map from \(\mu\) to \(\nu\), \[ D(\mu,\nu) = \int_S \left[ \int_{\mathbb R^d}|\theta\cdot(T(x)-x)|^2\,d\mu(x)-W_2^2(\mu_\theta,\nu_\theta) \right] d\sigma(\theta). \] This expresses the deficit as an average of nonnegative one-dimensional optimality gaps. In that form, the deficit becomes a rigidity functional rather than merely a difference of distances. The rigidity theorem states that \[ D(\mu,\nu)=0 \quad\Longleftrightarrow\quad T(x)=\lambda x+b \quad \mu\text{-a.e.} \] for some \(\lambda\ge 0\) and \(b\in\mathbb R^d\). Equality in the sliced inequality thus forces the Brenier map to be homothetic affine, not merely affine. The paper also proves a higher-dimensional Grassmannian analogue for \(k\)-dimensional projections. For quantitative stability, the paper introduces the ridge defect \[ R_\mu(u) := \int_S \mathbb E_\mu\!\left[\operatorname{Var}(\theta\cdot u(X)\mid \theta\cdot X)\right]\,d\sigma(\theta), \] and the sliced Poincaré–Korn constant \(\kappa_{\mathrm{SPK}}(\mu)\), whose null space is \[ A_d=\{x\mapsto \lambda x+b:\lambda\in\mathbb R,\ b\in\mathbb R^d\}. \] If \(\kappa_{\mathrm{SPK}}(\mu)>0\) and the projected monotone transports have uniformly bounded one-dimensional Lipschitz scale \[ \Lambda:=\operatorname*{ess\,sup}_{\theta\in S^{d-1}}\operatorname{Lip}(\tau_\theta)<\infty, \] then \[ \operatorname{dist}_{L^2(\mu)}^2(T,A_d) \le \frac{\Lambda}{\kappa_{\mathrm{SPK}}(\mu)}\,D(\mu,\nu). \] The paper emphasizes that \(\Lambda\) is a one-dimensional regularity scale; for Gaussian source measures and strongly log-concave targets, Caffarelli’s contraction theorem yields the uniform bound \(\operatorname{Lip}(\tau_\theta)\le \alpha^{-1/2}\). The Gaussian model gives the sharp benchmark: \[ \kappa_{\mathrm{SPK}}(\gamma_d)=\frac{d-1}{d(d+2)}, \qquad \overline\kappa_{\mathrm{SPK}}(\gamma_d)=\frac{d-1}{d+2}, \] for the standard Gaussian \(\gamma_d=N(0,I_d)\), and the same sharp constant holds for every isotropic Gaussian \(N(a,\sigma^2 I_d)\). Positive SPK bounds also persist under bounded \(L^\infty\) perturbations of the Gaussian: \[ d\mu=\rho\,d\gamma_d,\qquad 0