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 generative adversarial (GAN) methods at the map level. Neither yields a uniquely identified alignment map: GWOT returns only a coupling on the available samples, and GAN-based maps are determined only up to marginal matching. Modelling representations as Riemannian metric measure (mm-) spaces, we give sufficient conditions for the map to be unique and show they hold generically. We split recovery into two stages. In the coupling step, an injectivity condition on local distance distributions (LDDs), admitting a sample-level diagnostic, reduces GWOT to a convex linear program. In the map-fitting step, we establish finite-sample identifiability when the unpaired samples come from the same underlying data points, and asymptotic identifiability of the population alignment when they are drawn independently. The resulting Neural Gromov-Monge Alignment (NGMA) algorithm matches or improves on supervised baselines.