Transport-Aware Inference on Vector Bundles: When Is Parallel Transport Statistically Beneficial?

Many learning problems attach a vector to every point of a curved space: a velocity at each point of a sphere, a feature at each vertex of a mesh. The vectors at each location live in their own vector space, the fibre, and together the fibres form a vector bundle. Vectors in different fibres cannot be averaged directly; one must first decide how to carry a vector from one location to another. Geometric machine learning usually uses parallel transport, on the grounds that it is the geometrically canonical choice. We ask when it is also the statistically better one. We show that the error of replacing transport by a fixed reference rule depends on the rule, so it is not an intrinsic curvature effect: two standard rules on the sphere have biases of opposite sign. Curvature enters invariantly through holonomy, the rotation of a vector carried around a closed loop. An exact mean-squared-error identity shows that at low curvature a biased reference that contracts noise beats transport, and that for varying fields the answer depends on curvature relative to the field’s own variation. To make the comparison operational for incomplete data, we develop Bundle-GEM, a generalized EM procedure whose parameters live in moving fibres. In controlled experiments the predicted crossover matches the exact theory, a plug-in estimate of the identity picks the better estimator in 94% of cases, full Bundle-GEM on data with up to 80% missing fibre values comes within 0.006 of the true-parameter error where fixed references and global frames fail, and a surface without a global frame shows the predicted breakdown of coordinate baselines. On real global wind fields (NCEP/NCAR reanalysis), a gauge-equivariant network beats all non-transport baselines on every test field, and the plug-in identity picks the better local estimator at 98% of locations.