Identifiable alignment of unpaired representations

Stas Syrota, Albert Lopez I Serrano, Johanne Franck, Søren Hauberg

Aligning two models without paired data requires a distance that both share, yet existing methods use Euclidean or spherical geometry. This is either an arbitrary choice or imposed by their loss function. We show that, under a precise form of the manifold hypothesis, the pullback of the data metric gives such a distance for any two models of one data-generating process. This holds whether the models are encoders, decoders or both, and whatever their architecture, loss, prior or training data. The alignment map preserves this distance and the distribution of representations, and is identifiable if and only if the data distribution breaks every symmetry of the data manifold. We show that this symmetry breaking holds generically. Identifiability makes the alignment coupling the unique optimum of Gromov--Wasserstein and of its weighted combinations with a linear program. The linear program alone has the same optimum if almost every data point has a distinct distribution of distances. The coupling, and with it the alignment map, is recovered exactly from permuted samples, even without identifiability, and consistently from independent ones. Our method, Neural Gromov--Monge Alignment…