Barycentric Optimal Transport

• Barycentric optimal transport is a weak OT formulation that uses cost functions based on the conditional mean (barycenter) of target variables.
• Its dual formulation reveals a convex order structure, ensuring uniqueness via Brenier maps in quadratic settings.
• Applications span constrained map estimation in machine learning to robust economic matching, with efficient mirror descent algorithms for discrete measures.

The barycentric optimal transport problem is a class of weak optimal transport (WOT) problems characterized by cost functions that depend on the source and the barycenter (conditional mean) of the target variables under a chosen coupling. This formulation encompasses classical quadratic cost but generalizes to convex costs and finds applications in constrained transport map estimation, weak matching, robust barycenters, and aggregation models in economics and machine learning.

1. Mathematical Formulation and Static Characterization

Let $\mu \in \mathcal P(\mathcal X)$ and $\nu \in \mathcal P(\mathcal Y)$ be probability measures on compact, convex domains. For a coupling $\pi \in \Pi(\mu,\nu)$, with disintegration $\pi(dx,dy)=\mu(dx)\,\pi(dy|x)$, the barycentric map at $x$ is the conditional mean: $$b(\pi,x) = \int_{\mathcal Y} y\,\pi(dy|x) = \mathbb E_{y\sim\pi(\cdot|x)}[y].$$ Given a cost $c(x,z)$, convex in $z$, the barycentric WOT problem is: $$\mathsf{WOT}_\mathrm{bary}(\mu,\nu) = \inf_{\pi\in\Pi(\mu,\nu)} \int_{\mathcal X} c\bigl(x,\,b(\pi,x)\bigr)\,\mu(dx).$$ The most studied special case uses quadratic cost $c(x,z)=\|x-z\|^2$, yielding the quadratic barycentric optimal transport problem (Gozlan et al., 5 Sep 2025, Guo et al., 26 Nov 2025, Paty et al., 2022).

2. Dual Formulation and Convex Order Structure

Under the condition that $z\mapsto c(x,z)$ is convex and lower semicontinuous, the dual can be written as (Paty et al., 2022): $$\mathsf{WOT}_\mathrm{bary}(\mu,\nu) = \inf_{\psi \in \mathrm{ConvLip}(\mathrm{conv}(\mathcal Y))} \left\{ \int_{\mathcal X} Q_c[\psi](x)\,\mu(dx) + \int_{\mathcal Y}\psi(y)\,\nu(dy) \right\}$$ where $Q_c[\psi](x) = \sup_{z} \{c(x,z) - \psi(z)\}$ is the Hamiltonian (sup-transform).

In the quadratic case, a convex order projection emerges: the barycentric OT cost is equal to the minimal $W_2^2$ distance among all projections of $\mu$ onto measures $\eta$ dominated by $\nu$ in convex order (i.e., $\eta \preceq_c \nu$) (Guo et al., 26 Nov 2025, Gozlan et al., 5 Sep 2025): $$\overline T_2(\mu,\nu) = \inf_{\eta\preceq_c\nu} W_2^2(\mu,\eta).$$ The optimizer $\bar\eta$ is unique and given as the pushforward of $\mu$ under a 1-Lipschitz convex function (Brenier map).

3. Dynamic (Benamou–Brenier) and Martingale Analogues

The barycentric OT problem admits a dynamic formulation analogous to the Benamou–Brenier formula for classical OT, but only penalizing the drift component (Guo et al., 26 Nov 2025): $$\overline T_2(\mu,\nu) = \inf_{(X_t)} \mathbb E\left[ \int_0^1 |v_t|^2dt \right],$$ where processes $X_t$ satisfy $dX_t = v_tdt + \sigma_t dB_t$, $X_0 \sim \mu$, $X_1 \sim \nu$, and $\sigma_t$ is arbitrary (martingale part is cost-free). This focuses transport cost on movement of conditional expectations, with extra noise/martingales unconstrained.

An interpolation between barycentric and martingale cost yields the $\alpha$–$\beta$ functional (Guo et al., 26 Nov 2025): $$\inf_{\pi\in\Pi(\mu,\nu)} \int [\alpha|x-\mathbb E[Y|X=x]|^2 - \beta\,\mathsf{MCov}(\pi_x, \gamma^d_1)]\,\mu(dx)$$ where $\mathsf{MCov}(\rho,\sigma)$ is the maximal covariance between measures $\rho$ and $\sigma$, and $\gamma^d_1$ is standard Gaussian.

In one dimension and for Gaussians, explicit formulae in terms of convex envelopes or linear SDEs are available (Gozlan et al., 5 Sep 2025, Guo et al., 26 Nov 2025).

4. Numerical Algorithms: Mirror Descent and Barycentric Projections

For discrete or sampled measures, first-order primal and dual mirror descent algorithms with KL (entropic) geometry are efficient in high dimensions (Paty et al., 2022, Deb et al., 2021). The primal maximizes over the coupling matrix $P$ with objective: $$f(P) = \sum_{i=1}^n a_i\, c\left(x_i,\, \sum_{j=1}^m P_{ij}y_j\right)$$ using gradient (in $P$) followed by KL-exponentiation and Sinkhorn projection to enforce marginals.

The dual works with convex potentials (parametrized on support) using subgradients derived from the Hamiltonian optimizer $z^*$.

Complexity per iteration is $O(nm)$ with convergence rate $O(1/\sqrt{T})$ for general convex cost and $O(1/T)$ if smooth/strongly convex. For quadratic cost, the barycentric projection of an optimal plan provides a plug-in estimator for maps, with nonparametric and smoothed estimators’ rates analyzed in (Deb et al., 2021).

In the context of constrained map estimation, the barycentric mapping for a plan $\pi$ provides the conditional mean $\bar\pi(x) = \mathbb{E}[Y|X=x]$, which is then projected onto a given function class $G$ (e.g., Lipschitz gradients, RKHS, neural nets) via $L^2$-minimization (Tanguy et al., 18 Jul 2024).

5. Theoretical Guarantees: Existence, Uniqueness, and Structure

• Existence of a barycentric OT optimizer is ensured by compactness and lower semicontinuity of the cost.
• In the quadratic barycentric setting, the optimizer splits as: Brenier map $\mu \to \bar\mu$ (convex order minimal measure) followed by any martingale coupling $\bar\mu \to \nu$, as characterized by Strassen-type theorems (Gozlan et al., 5 Sep 2025).
• Dual optimizers exist as convex potentials, providing a global duality (Paty et al., 2022, Gozlan et al., 5 Sep 2025).
• In dimension one, explicit solutions exist in terms of quantile functions (Guo et al., 26 Nov 2025).

6. Applications Across Disciplines

• Economics: Models of labor market matching with aggregation effects use barycentric WOT for capturing production via team composition (firm as $x$, worker types $y$, cost is output as CES function on barycenters) (Paty et al., 2022).
• Machine Learning: Robust barycenters, low-regularity OT map estimation, and adversarial or continuous relaxations of statistical independence in generative modeling employ barycentric projection and related weak OT formulations (Tanguy et al., 18 Jul 2024, Deb et al., 2021).
• Statistics: Barycentric projections underpin plug-in estimators for transport maps and test statistics for independence (Deb et al., 2021).
• Numerical Linear Programming: Fast combinatorial algorithms for the barycentric problem and its variants leverage the structure of barycentric costs and sparsity of supports (Borgwardt, 2017, Anderes et al., 2015).

7. Connection to Classical OT and Beyond

• The barycentric OT is a generalization of classical OT; it recovers Monge-Kantorovich when the cost is linear in $y$.
• Its distinguishing feature is cost nonlinear in the conditional distribution, only convex in barycenters, which restricts original OT plans but provides a generalized matching framework.
• Its dynamic and martingale analogues connect optimal transport to stochastic processes, Markov representations, and more general convex geometries (Guo et al., 26 Nov 2025, Gozlan et al., 5 Sep 2025).
• Variants include density-dependent minimal action (Lagrangian-weighted costs) (Wang et al., 4 Nov 2025), unbalanced extensions for robust barycenters (Nguyen et al., 10 Oct 2024), and constrained or regularized map estimation through projection (Tanguy et al., 18 Jul 2024).

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References:

• (Paty et al., 2022) Algorithms for Weak Optimal Transport with an Application to Economics
• (Guo et al., 26 Nov 2025) Dynamic characterization of barycentric optimal transport problems and their martingale relaxation
• (Gozlan et al., 5 Sep 2025) On the quadratic barycentric transport problem
• (Tanguy et al., 18 Jul 2024) Constrained Approximate Optimal Transport Maps
• (Deb et al., 2021) Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections
• (Borgwardt, 2017) An LP-based, Strongly-Polynomial 2-Approximation Algorithm for Sparse Wasserstein Barycenters
• (Anderes et al., 2015) Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data