Barycentric Transport Methods

• Barycentric Transport is an optimal transport framework that uses conditional barycenters or Fréchet means to construct deterministic mappings from probabilistic couplings.
• It underpins efficient algorithms in high-dimensional problems, leveraging grid-based LP solvers and intrinsic projections on both Euclidean and Riemannian manifolds.
• Its applications extend to barycenter computation, partial transport, dynamic matching, and distributed data fusion, backed by strong statistical stability and convergence results.

Barycentric transport refers to a family of optimal transport (OT) methods and formulations where conditional barycenters—typically, conditional expectations or Fréchet means—play a key role in constructing transport maps, projections, or barycenters of probability measures. The barycentric projection is formally the assignment of each source point to the expectation of its conditional target under a coupling, and this principle generalizes to Riemannian manifolds, non-Euclidean geometries, and a variety of weak OT settings. Barycentric transport underpins both efficient algorithms for high-dimensional OT and the definition and computation of Wasserstein barycenters, and is central in formulating dynamic, causal, and multicausal optimal transport.

1. Mathematical Formulation of Barycentric Projection

Let $\mu,\nu$ be measures on compact sets $X,Y\subset\mathbb{R}^d$, and $\pi\in\Pi(\mu,\nu)$ an optimal transport plan (i.e., solution to the Kantorovich problem). By disintegration, one writes

$\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$

where $\pi_x\in\mathcal{P}(Y)$ is a regular conditional probability. The barycentric projection $T: X \to Y$ is defined by

$T(x) = \int_Y y \, d\pi_x(y)$

for $\mu$-almost every $x$. For discrete measures $\mu^h = \sum_i \mu^h_i \delta_{x_i}$, $X,Y\subset\mathbb{R}^d$0, and plan $X,Y\subset\mathbb{R}^d$1, the barycentric projection is

$X,Y\subset\mathbb{R}^d$2

assigning $X,Y\subset\mathbb{R}^d$3 (Oberman et al., 2015).

This construction is canonical in the sense that, when the optimal plan is induced by a deterministic Monge map $X,Y\subset\mathbb{R}^d$4, the barycentric projection recovers $X,Y\subset\mathbb{R}^d$5 $X,Y\subset\mathbb{R}^d$6-a.e.

2. Barycentric Projection in Algorithmic OT and Barycenter Computation

Barycentric projection enables efficient computation of deterministic maps from probabilistic couplings, thus bridging Kantorovich and Monge formulations. In the context of grid-based linear programming (LP) OT solvers, Oberman–Ruan demonstrate a multiscale refinement method where the LP solution at each grid scale is post-processed by barycentric projection to produce high-resolution, approximately optimal maps. The key steps are:

• Discretize $X,Y\subset\mathbb{R}^d$7 and $X,Y\subset\mathbb{R}^d$8 on a grid.
• Solve the LP for the transport plan.
• For each source grid point $X,Y\subset\mathbb{R}^d$9, compute the barycentric projection $\pi\in\Pi(\mu,\nu)$0.
• Iterate on refined grids to improve resolution and localization.

This approach reduces the computational cost of extracting transport maps to $\pi\in\Pi(\mu,\nu)$1—negligible compared to the LP solve—and converges weakly to the Monge solution under convexity and regularity assumptions (Oberman et al., 2015). It extends naturally to partial OT (with inequality constraints) and to barycenter computations: after solving for a barycenter $\pi\in\Pi(\mu,\nu)$2 and associated coupling plans $\pi\in\Pi(\mu,\nu)$3, one barycentrically projects each $\pi\in\Pi(\mu,\nu)$4 to obtain maps from $\pi\in\Pi(\mu,\nu)$5 to each $\pi\in\Pi(\mu,\nu)$6.

3. Barycentric Projections on Manifolds and Intrinsic Approaches

On Riemannian manifolds $\pi\in\Pi(\mu,\nu)$7, the notion of barycentric projection generalizes to the assignment of each $\pi\in\Pi(\mu,\nu)$8 to the conditional Fréchet mean of $\pi\in\Pi(\mu,\nu)$9, i.e.,

$\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$0

with $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$1 the geodesic distance. The intrinsic barycentric projection yields the best deterministic representative of the coupling under squared geodesic loss and uniquely recovers map-induced couplings (i.e., Monge maps). The tangential log-exp projection assigns $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$2 to $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$3, acting as a local, first-order surrogate. These projections characterize the Monge defect—vanishing only if the coupling is induced by a map—and are supported by strong variational optimality and statistical stability results (You, 6 Jun 2026).

4. Statistical Properties and Rates for Barycentric Plug-in Estimators

Barycentric projection is fundamental in statistical estimation of OT maps. For empirical measures $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$4 and $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$5 supported on finite samples, the barycentric projection of any optimal $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$6 is

$\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$7

When the true Monge map $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$8 exists, the plug-in barycentric projection converges to $\pi(dx,dy) = \pi_x(dy) \,\mu(dx)$9. Deb–Ghosal–Sen establish stability estimates and rates: $\pi_x\in\mathcal{P}(Y)$0 and show that smoothing (via wavelets or kernels) can accelerate convergence, approaching the minimax optimal rate under sufficient regularity (Deb et al., 2021).

5. Barycentric Transport in Weak and Barycentric OT Problems

Barycentric (or weak) transport costs, as developed by Gozlan, Juillet, and others, measure the mean squared displacement of conditional barycenters: $\pi_x\in\mathcal{P}(Y)$1 This problem, also known as WOT$\pi_x\in\mathcal{P}(Y)$2, is fundamentally barycentric in nature. The optimizer arises via a "backward" projection of $\pi_x\in\mathcal{P}(Y)$3 under a 1-Lipschitz Brenier-type map, followed by a martingale coupling to $\pi_x\in\mathcal{P}(Y)$4. There is an explicit duality, as well as a Benamou--Brenier dynamic characterization in terms of minimal kinetic energy over drifted (possibly noise-driven) processes (Gozlan et al., 5 Sep 2025, Guo et al., 26 Nov 2025).

The relation

$\pi_x\in\mathcal{P}(Y)$5

shows that barycentric cost is a projection under convex order, encoding a weaker constraint than standard OT but still exhibiting deep geometric structure.

6. Fixed-Point and Iterative Schemes for Barycenters and Maps

For general cost functions, barycentric projection is central in practical fixed-point algorithms for free-support or generic-cost barycenters. The Tanguy–Delon–Gozlan fixed-point map

$\pi_x\in\mathcal{P}(Y)$6

with $\pi_x\in\mathcal{P}(Y)$7 allows rapid, flexible computation in challenging settings, converging under compactness and uniqueness assumptions (Tanguy et al., 2024). On manifolds and in multiscale LP solvers (Oberman et al., 2015), the barycentric projection is used at each iteration to extract an approximate transport map.

In Gromov–Wasserstein theory, barycentric transport structures the tangent-space linearization and embedding construction, underpinning theoretically grounded algorithms for the computation of GW barycenters and shape interpolations (Beier et al., 2024).

7. Applications and Generalizations

Barycentric transport underlies a wide range of OT and barycenter methods:

• Partial OT and Free-Boundary Problems: Barycentric projection identifies the region of the source actually transported in partial transport, accurately resolving free boundaries (Oberman et al., 2015).
• Geometric Data Fusion: In time-frequency super-resolution, barycenter fusion combines diverse localizations, with the barycentric step yielding the optimal averaging of energies across heterogeneous grids (Valdivia et al., 16 Apr 2026).
• Dendritic/Axially-Oriented Measures: In layered or treelike geometries, the barycentric structure ensures the preservation of biological or physical constraints (e.g., root systems), via decoupling into layerwise barycenters (Kim et al., 2019).
• Federated or Distributed Barycenter Computation: Dual decomposition algorithms efficiently select barycenter support points relying only on aggregated barycentric quantities, ensuring privacy and scalability (Lin et al., 25 Jul 2025).
• Dynamic Matching and Causal Transport: Barycenter principles extend to multicausal or adapted stochastic process settings, where the causal barycenter is a variational equilibrium against adapted couplings (Acciaio et al., 2024).

These diverse settings showcase the versatility and foundational character of barycentric transport in both the theory and the computation of optimal transport and its generalizations.

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Key References

• (Oberman et al., 2015)—Efficient LP barycentric projection, convergence, and applications to partial/barycenter transport.
• (You, 6 Jun 2026)—Intrinsic and tangential barycentric projections on Riemannian manifolds.
• (Deb et al., 2021)—Statistical properties, stability, and rates for barycentric plug-in estimators.
• (Gozlan et al., 5 Sep 2025, Guo et al., 26 Nov 2025)—Quadratic barycentric transport and dynamic formulations.
• (Tanguy et al., 2024)—Fixed-point-based barycentric computation for generic costs.
• (Beier et al., 2024)—Barycentric transport in GW barycenters.
• (Kim et al., 2019)—Layerwise barycentric OT for dendritic measures.
• (Lin et al., 25 Jul 2025)—Federated dual algorithms for barycenter selection and transport.