Kisung You's Framework Converts Optimal Transport to Deterministic Maps on Curved Spaces

Optimal transport couplings are inherently probabilistic, but many machine learning pipelines require deterministic maps. In Euclidean space, barycentric projection (taking conditional expectations) solves this, but curved spaces (Riemannian manifolds) introduce curvature and cut loci that complicate the operation. Kisung You's paper develops a rigorous framework for barycentric projections on manifolds, offering two key constructions. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination distribution, and is proven to be the best deterministic representative under squared geodesic loss. Its error—an integrated conditional Fréchet variance—vanishes exactly for map-induced couplings, defining a 'conditional-variance Monge defect' that quantifies how far a coupling is from being a deterministic map.

The second construction is a tangential log-exp projection that uses local linearization via the logarithm map. This projection is exact in Euclidean spaces, compatible with Brenier-McCann maps in the Monge case, and can be interpreted as the first Riemannian gradient step toward the intrinsic optimum. For discrete couplings, both methods decompose row-wise into weighted Fréchet mean and log-exp problems, making them computationally tractable. Experiments on spherical data, synthetic symmetric positive-definite (SPD) matrices, and real EEG covariance matrices demonstrate a clear division of roles: the intrinsic projection serves as the true variational representative, while the tangential projection acts as a useful local displacement surrogate. This work bridges optimal transport theory and practical geometric machine learning.