Learning Across Feature Dimensions via Dimension-Conditioned Latent Schrödinger Transport

We study supervised learning when observations of the same type are represented by different numbers of features. The training data are grouped by feature dimension, and cross-dimensional sample identities are not used. The objective is a single predictor that operates on all training dimensions and transfers to dimensions not used for fitting. We introduce a dimension-conditioned encoder followed by a conditional latent transport to a common reference distribution. The transport is formulated as a soft-constrained multi-marginal Schrödinger bridge and approximated by conditional flow matching. A single task head is then trained on the transported representations. We give a uniform score-error decomposition, a dimension-coverage bound under an assumed per-dimension estimation rate, and a penalty-convergence result for the ideal soft bridge. Synthetic experiments examine shared prediction, dimension coverage, and finite-network penalty behavior. In an independent PBMC3k benchmark retraining, the final configuration attains mean accuracy $0.811$, macro-F1 $0.683$, and macro-AUROC $0.947$ across six feature dimensions. Dimension-specific classifiers remain stronger in accuracy and F1, which identifies the present cost of using one model across dimensions.