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.

翻译：最优传输耦合是概率性对象，而许多学习流程需要确定性映射。在欧氏空间中，重心投影通过条件期望将耦合转化为映射，但在黎曼流形上，曲率和切割轨迹使得这一操作变得复杂。我们发展了一个在黎曼流形上对传输耦合进行重心投影的框架。内在投影将每个源点映射到其目标律的条件Fréchet均值，并证明其是平方测地线损失下的最佳确定性代表。对应的最小值为一个条件Fréchet方差，该方差在由映射诱导的耦合中恰好为零，因此定义了一个条件方差Monge缺陷。我们还研究了切向对数-指数投影，证明了它的欧氏精确性、在Monge情形下与Brenier-McCann映射的兼容性，以及它对于内在目标函数作为第一个单位黎曼梯度更新的解释。对于离散耦合，两种构造均按行分解为加权Fréchet均值和对数-指数问题。在球面数据、合成SPD数据和真实EEG协方差矩阵上的实验支持了所提出的角色分工：内在投影是变分代表，而切向投影是有用的局部位移代理。