Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

Abstract: Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss. The corresponding minimum value is an integrated conditional Fréchet variance, which vanishes exactly for map-induced couplings and therefore defines a conditional-variance Monge defect. We also study a tangential log-exp projection, prove its Euclidean exactness, its compatibility with Brenier-McCann maps in the Monge case, and its interpretation as the first unit Riemannian gradient update for the intrinsic objective. For discrete couplings, both constructions decompose row-wise into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices support the proposed division of roles: the intrinsic projection is the variational representative, while the tangential projection is a useful local displacement surrogate.

• The paper introduces intrinsic and tangential barycentric projections to convert probabilistic OT couplings into deterministic maps on Riemannian manifolds while addressing curvature challenges.
• It presents a variational framework for the intrinsic projection and a log–exp averaging strategy for the tangential projection, highlighting their equivalence and limitations.
• Experimental validations on spheres, SPD matrices, and EEG data demonstrate the practical effectiveness and stability of the proposed methods.

Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

Introduction and Problem Statement

The study addresses the conversion of optimal transport (OT) plans—probabilistic couplings—into deterministic maps on Riemannian manifolds. While barycentric projection provides this transition in Euclidean space via conditional expectations, the generalization to curved, geodesic spaces presents substantial difficulties due to cut loci and curvature. The work presents a principled framework for defining and computing barycentric projections of transport plans on Riemannian manifolds, clarifying their geometric and variational significance, and operationalizing them for practical manifold-valued machine learning tasks.

The central objects are:

• Intrinsic barycentric projection: Defined by the conditional Fréchet mean with respect to the geodesic distance.
• Tangential barycentric projection: Defined by averaging logarithmic displacement vectors in the tangent space at each source, with the result mapped back via the exponential map. These constructions represent fundamentally different geometric mechanisms in the non-Euclidean setting. The intrinsic projection solves a variational characterization, while the tangential projection yields a practical, first-order surrogate closely linked to displacement interpolation and gradient-based updates.

Mathematical Framework and Key Results

Geometry and OT Couplings

Given a complete, connected finite-dimensional Riemannian manifold $(\mathcal{M}, g)$ and two probability measures $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$, the authors consider couplings $\pi \in \Pi(\mu, \nu)$ with finite quadratic cost $c(x,y) = \frac{1}{2} d(x,y)^2$. The coupling may (and typically does) split the mass of a source atom to multiple destinations, especially for empirical measures with unequal support sizes.

A crucial conceptual shift is viewing the barycentric projection as a geometric plan-to-map operation—not just as numerical postprocessing—since the structure of OT couplings in curved spaces is fundamentally non-deterministic.

Intrinsic Barycentric Projection

The intrinsic projection at point $x$ is the minimizer of the conditional Fréchet functional:

$$F_x(z) = \frac{1}{2}\int_\mathcal{M} d(z,y)^2\,d\pi(y|x),$$

where $\pi(y|x)$ is the kernel from the disintegration of $\pi$ along $\mu$. Existence, uniqueness (under curvature assumptions or local convexity), Borel regularity, and integrability of these minimizers are formally established. The resulting projection $B_\pi(x)$ is shown to be the unique, optimal deterministic map under expected squared geodesic loss.

A key structural result is that the minimum value $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$0 of this objective across all Borel deterministic maps equals the conditional-variance Monge defect. This defect vanishes if and only if the OT plan is Monge-induced (i.e., deterministic), and is thus a robust measure of the failure of the coupling to be induced by a map.

Monge Consistency: If the OT plan is already map-induced, the intrinsic projection recovers the original map almost everywhere.

Tangential Barycentric Projection

The tangential projection is defined where the logarithmic map is single-valued and integrable:

$\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$1

This is interpreted as averaging displacement vectors in the tangent space $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$2 and mapping back through the exponential. The construction is Borel measurable (in the Hadamard regime) and strictly coincides with the intrinsic projection in the Euclidean case. In the Monge setting, it recovers the Brenier--McCann displacement, i.e., the vector field whose exponential is the optimal map. In the general case, it constitutes a first-order update for the intrinsic Fréchet mean objective:

$\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$3

so $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$4 is the result of a unit step Riemannian gradient update starting from $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$5.

Limitations: If the conditional destinations concentrate on the cut locus, the tangential projection may be undefined, even when the intrinsic mean exists.

Discrete and Algorithmic Aspects

For empirical measures and discrete couplings, both projections reduce to parallel, row-wise operations:

• The intrinsic projection is a (possibly iterative) weighted Fréchet mean computation for each source atom.
• The tangential projection is a single log–exp average per row. A hybrid algorithm is suggested: use the tangential projection to initialize Fréchet mean solvers, which maintains the variational property while often improving computational efficiency.

Stability: The solutions are continuous with respect to perturbations of the coupling weights, established via elementary continuity and measurable selection arguments.

Experimental Validation

Experiments demonstrate the theory and practical distinctions on three types of data: $\mu,\,\nu \in \mathcal{P}_2(\mathcal{M})$6 (sphere), SPD(3) matrices (affine-invariant metric, Hadamard setting), and real EEG covariance data.

• Intrinsic projections always achieve minimum plan-to-map energy (the Monge defect), validating the variational theorem across regimes.
• Tangential projections are extremely close to intrinsic ones in Hadamard settings and in low-dispersion rows, highlighting the local linearization quality of the tangent approximation.
• On configurations with high curvature or large support separation (sphere), discrepancies are visible and highlight the surrogate nature of the log–exp projection.

Energy-gap inequalities in the Hadamard regime are observed to be numerically sharp. Target mismatch is empirically controlled by the Monge defect as predicted.

Implications and Future Directions

This framework clarifies the geometry and mechanics of extracting deterministic correspondences from optimal transport couplings on general manifolds. Practically, it enhances manifold-domain learning tasks—including alignment, adaptation, and geometric domain translation—by selecting optimal deterministic surrogates for probabilistic transport structures. The conditional-variance Monge defect provides a rigorous measure of the gap between probabilistic and deterministic representations, informing both theoretical understanding and empirical model selection.

Theoretically, the results reinforce the significance of Riemannian convexity and curvature in algorithmic OT, and suggest new approaches for interpolating between probabilistic and deterministic learning frameworks on manifolds. Algorithmically, the decoupling into row-wise problems and hybrid initializations open pathways to scalable implementations in high-dimensional geometric machine learning.

Potential future directions include:

• Extension to entropic/smooth OT (involving more complex couplings and dual potentials).
• Broader families of cost functions (beyond the squared geodesic).
• Investigation of the statistical properties of the barycentric selection for high-dimensional or infinite-dimensional manifolds.
• Downstream applications in geometry-aware neural networks, domain adaptation, and non-Euclidean generative modeling.

Conclusion

The paper establishes a comprehensive theory and practical methodology for barycentric projections of optimal transport plans on Riemannian manifolds, distinguishing intrinsic variational representatives from tangential displacement surrogates. The developed results have direct implications for manifold-valued geometric learning and reveal sharp characterizations of when and how probabilistic transport can be optimally collapsed to deterministic maps, clarifying foundational connections between optimal transport, averaging, and Riemannian geometry (2606.07926).