Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs

The ability to compare and align related datasets living in heterogeneous spaces plays an increasingly important role in machine learning. The Gromov-Wasserstein (GW) formalism can help tackle this problem. Its main goal is to seek an assignment (more generally a coupling matrix) that can register points across otherwise incomparable datasets. As a non-convex and quadratic generalization of optimal transport (OT), GW is NP-hard. Yet, heuristics are known to work reasonably well in practice, the state of the art approach being to solve a sequence of nested regularized OT problems. While popular, that heuristic remains too costly to scale, with cubic complexity in the number of samples $n$. We show in this paper how a recent variant of the Sinkhorn algorithm can substantially speed up the resolution of GW. That variant restricts the set of admissible couplings to those admitting a low rank factorization as the product of two sub-couplings. By updating alternatively each sub-coupling, our algorithm computes a stationary point of the problem in quadratic time with respect to the number of samples. When cost matrices have themselves low rank, our algorithm has time complexity $\mathcal{O}(n)$. We demonstrate the efficiency of our method on simulated and real data.