Identifiable alignment of unpaired representations

Aligning representations drawn from a shared distribution without prior correspondence is typically approached through Gromov-Wasserstein Optimal Transport (GWOT) at the coupling level, or through Neural OT and generative adversarial (GAN) methods at the map level. None yields a uniquely identified alignment map, since GWOT returns only a coupling on the available samples, Neural OT maps target a user-chosen ground cost, and GAN-based maps are determined only up to marginal matching. Modelling representations as Riemannian metric measure (mm-) spaces under the identifiable pullback geometry of their decoders, we give sufficient conditions for the map to be unique and show they hold for generic measures. We split recovery into two stages. In the coupling stage, an injectivity condition on local distance distributions (LDDs) reduces GWOT to a convex linear program, and we quantify the condition from samples through a margin with a DKW-based certificate. Recovery is exact at finite $n$ when the unpaired samples share underlying data points; when they are drawn independently, we bound the coupling mass misplaced at every finite $n$ and establish asymptotic identifiability of the population alignment. The resulting _Neural Gromov-Monge Alignment (NGMA)_ algorithm matches fully supervised baselines.