Conditional Gromov–Wasserstein matching across contexts: theory, regularization, and applications to brain connectivity

The Gromov–Wasserstein (GW) distance is a principled framework for comparing structured data when pointwise correspondences are unknown, but standard formulations assume that each object has a fixed geometry. This assumption is restrictive when relational structure changes with context, such as cognitive state, task condition, or experimental environment. We propose Conditional Gromov–Wasserstein (CGW), a context-aware extension of GW for comparing metric measure spaces whose internal geometry depends on a conditioning variable. The framework comprises three related formulations: an expected formulation that learns a single coupling shared across contexts, a conditional formulation that learns a distinct coupling for each context, and a regularized formulation that encourages smooth variation of couplings across contexts. We establish the existence of minimizers for the discrete formulations and show stability under perturbations of the context-dependent cost matrices. We also provide a variational interpretation of the regularized model through proximal transport and its connection to the Jordan–Kinderlehrer–Otto scheme and, under entropic regularization, a Schrödinger-bridge interpretation of the resulting family of couplings. Synthetic experiments, using connectivity-style graphs inspired by rest- versus task designs, show that the proposed formulations recover a known, constructed pattern of stable structure plus localized rewiring, without using real brain data. In particular, the conditional and regularized variants produce smoother alignments than classical GW and allow systematic control of coupling variation through the regularization parameter. These results suggest that CGW is a useful framework for modeling relational data in which shared structure and context-specific adaptation coexist.