Barycentric Projections in Optimal Transport

• Barycentric projections in optimal transport are mappings that assign each source point a representative barycenter of its conditional distribution, encapsulating key deterministic and stochastic features.
• They underpin dual formulations and martingale decompositions, simplifying the analysis of both classical and weak transport problems with convex cost functions.
• Numerical methods, including entropic regularization, use barycentric projections to derive continuous approximations from discrete plans, boosting efficiency in high-dimensional settings.

Barycentric projections of optimal transport plans refer to the construction, analysis, and implementation of mappings that assign to each point in the source space a representative “barycenter”—often a mean or centroid—of the conditional distribution over target points specified by an optimal (or weakly optimal) transport plan. Barycentric projections arise in classical optimal transport, weak transport problems, and entropic regularizations, and serve as a unifying tool in both the duality theory and computational practice of transport.

1. Mathematical Definition and Conceptual Framework

Given probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$ (or more generally, on a Polish space) and a transport plan $\pi \in \Pi(\mu,\nu)$, the barycentric projection $T_\pi$ is defined as: $T_\pi(x) = \int y\, d\pi^x(y)$ where $\pi^x$ is the disintegration ("conditional law") of $\pi$ given $x$ (see (Pramenković, 9 Jul 2025, Gozlan et al., 5 Sep 2025, Benamou et al., 2014, Oberman et al., 2015)). In applications where $\pi^x$ is not a Dirac (as in the Monge case), $T_\pi$ gives a canonical summary: the barycenter (expected value) of the assigned mass.

In weak optimal transport and its extensions (e.g., barycentric or quadratic barycentric transport), the cost depends explicitly on the mean (or moments) of $\pi^x$ rather than on specific realizations $y$ (see (Gozlan et al., 2018, Pramenković, 9 Jul 2025, Gozlan et al., 5 Sep 2025)).

2. Barycentric Projections in Weak and Barycentric Optimal Transport

In weak OT, the cost $C(x,\rho)$ is a function of $x$ and an entire conditional measure $\rho \in \mathcal{P}_1(\mathbb{R}^d)$, often through its barycenter: $C(x, \rho) = \theta(x - \text{mean}(\rho))$ for a convex function $\theta$ (Pramenković, 9 Jul 2025, Gozlan et al., 5 Sep 2025, Gozlan et al., 2018). This leads to defining the "barycentric projection" of the optimal plan as the map minimizing the expected target mean conditioned on $x$. In the quadratic barycentric case: $$T_2(\nu|\mu) = \inf_{\pi\in\Pi(\mu,\nu)} \int |E[Y|X] - X|^2 d\mu(x)$$

A key finding is the decomposition of optimal plans into a deterministic map (the barycentric projection) followed by a martingale coupling (see (Gozlan et al., 5 Sep 2025, Gozlan et al., 2018)):

• The map $x \mapsto \nabla \varphi(x)$ yields the measure $\bar{\mu} = (\nabla \varphi)_\# \mu$ which is the projection of $\mu$ onto measures less than $\nu$ in convex order.
• The complete optimal plan is $(X,Y)$ with $X \sim \mu$, first mapped deterministically via $X \mapsto \nabla \varphi(X)$, then a martingale kernel sends $\bar{\mu}$ to $\nu$.
• The martingale condition $E[Y|X] = T_\pi(X) = \nabla\varphi(X)$ characterizes barycentric optimality.

In full generality, barycentric projections allow the restriction of dual potentials to convex functions, giving dual forms where

$$\sup_{\psi\ \text{convex}} \left[ \mu(\psi^c) - \nu(\psi) \right]$$

with $\psi^c$ the $C$-conjugate defined via the barycentric cost (Pramenković, 9 Jul 2025, Gozlan et al., 2018, Gozlan et al., 5 Sep 2025).

3. Dual Formulations and Structure of Optimal Plans

Duality is central to the theory and effectiveness of barycentric projections:

• In quadratic barycentric transport (Gozlan et al., 5 Sep 2025), the Kantorovich dual problems use operators $Q_2$ and $P_2$:

$$Q_2 f(x) = \inf_{y\in\mathbb{R}^d}\left\{f(y) + \frac12|x-y|^2\right\}$$

$$P_2 g(y) = \sup_{x\in\mathbb{R}^d}\left\{g(x) - \frac12|x-y|^2\right\}$$

• The barycentric projection $V_y(x)$ (arising from the dual optimizer) is 1-Lipschitz, and its pushforward defines the "backward projection" $p^* = (V_y)_\# p$.
• The optimal plan $\pi$ decomposes as $d\pi(x, y) = dp(x) \cdot dq_{V_y(x)}(y)$: deterministic mapping (projection) followed by a martingale transport.
• This structure gives rise to interpolation formulas, such as in geodesics of the barycentric transport problem (Gozlan et al., 5 Sep 2025).

4. Computation and Algorithms

Numerical approaches employing barycentric projections are prolific:

• Entropic regularization and iterative Bregman projections (Benamou et al., 2014) allow reformulating OT problems as KL projection problems. Here, the barycentric projection arises as the solution to iterative alternate projections, e.g., Sinkhorn-like scaling for barycenter computation.
• In discrete or high-dimensional contexts, barycentric projections are crucial for extracting meaningful maps:
  • Barycentric projection as post-processing of an LP solution via $\bar{y}_i = \sum_j \pi_{ij} y_j / \sum_j \pi_{ij}$ (Oberman et al., 2015), yielding convergence to continuous optimal maps as the grid is refined.
  • Plug-in estimators for empirical OT maps are constructed via barycentric projections, with established statistical rates and stability properties (Deb et al., 2021, Pooladian et al., 2021, Mordant, 16 Dec 2024).
• In weak transport and barycentric WOT algorithms (Paty et al., 2022), primal and dual mirror descent iterations exploit the barycentric cost structure, resulting in improved numerical behaviors and interpretability.

5. Connections to Barycenters and Multi-marginal OT

Barycentric projections underpin the computation of Wasserstein barycenters:

• In Wasserstein barycenter problems, the barycenter measure serves as a fixed point under the barycentric projection operator defined by averaging the pushforwards of the optimal maps from the candidate barycenter to each marginal (Álvarez-Esteban et al., 2015, Tanguy et al., 20 Dec 2024, Visentin et al., 28 May 2025).
• The connection extends to multi-marginal transport; the barycentric projection yields, through a fixed-point or gradient-based procedure, the barycenter for arbitrary cost functions across families of measures (Tanguy et al., 20 Dec 2024).
• In discrete settings, robust non-mass-splitting barycenter structure and sparse supports are achieved by construction of LPs guided by barycentric projections (Anderes et al., 2015, Borgwardt, 2017).
• In sliced and expected sliced transport, the barycentric projection is synthesized by lifting 1D optimal plans and averaging; this enables metrics and embeddings in high-dimensions (Liu et al., 16 Oct 2024, Muzellec et al., 2019).

6. Geometric, Theoretical, and Practical Implications

Barycentric projections offer significant geometric and structural advantages:

• They encode convex ordering: crucial in martingale OT, they preserve convex order constraints, and the Markov property in stochastic processes arising from barycentric interpolations (Gozlan et al., 5 Sep 2025).
• In weak optimal transport, restricting the dual to convex potentials dramatically simplifies theory and computation, unifying classical, barycentric, and martingale transport frameworks (Pramenković, 9 Jul 2025, Gozlan et al., 5 Sep 2025, Gozlan et al., 2018, Paty et al., 2022).
• In applications ranging from economics (labor market matching, utility models), image processing, diffusion models, and generative modeling, barycentric projections offer interpretable and statistically robust mappings and summaries (Paty et al., 2022, Visentin et al., 28 May 2025, Mordant, 16 Dec 2024).
• Statistically, entropy-regularized barycentric projections yield estimators for the score function, with central limit theorems characterizing their asymptotic fluctuations as the regularization parameter shrinks (Mordant, 16 Dec 2024).

7. Summary of Main Formulas and Properties

Name	Formula/Property	Reference
Barycentric projection	$T_\pi(x) = \int y\, d\pi^x(y)$	(Oberman et al., 2015, Benamou et al., 2014)
Quadratic bary. cost	$T_2(v|p) = \inf_\pi E[|E[Y|X] - X|^2]$	(Gozlan et al., 5 Sep 2025, Gozlan et al., 2018)
Dual with $Q_2$	$Q_2f(x) = \inf_{y}\{f(y) + \frac12|x - y|^2\}$	(Gozlan et al., 5 Sep 2025, Gozlan et al., 2018)
Structure of optimal	$d\pi(x, y) = dp(x) \cdot dq_{V_y(x)}(y)$; $V_y$ is 1-Lipschitz barycentric map	(Gozlan et al., 5 Sep 2025)
Benamou-Brenier analog	$T_2(v|p) = \inf E[\int_0^1 |v_t|^2 dt]$, $X_t = (1-t)X_0+t V_y(X_0)+M_t-V_y(X_0)$	(Gozlan et al., 5 Sep 2025)
Weak OT dual	$$\sup_{\psi~\text{convex}} [\mu(\psi^c) - \nu(\psi)]$$	(Pramenković, 9 Jul 2025, Gozlan et al., 2018)

Barycentric projections thus serve as a conceptual and computational cornerstone for optimal transport problems, streamlining duality, informing algorithm design, and providing theoretical and practical insights in both deterministic and weak/martingale frameworks. The decomposition into a canonical barycentric mapping and (when present) a martingale or further “noise” part underpins much of the modern structure theory of transport and its applications in statistics, high-dimensional geometry, and economics.