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

• URL: http://arxiv.org/abs/2106.01128v1
• Date: Wed, 2 Jun 2021 12:50:56 GMT
• Title: Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs
• Authors: Meyer Scetbon, Gabriel Peyr\'e, Marco Cuturi
• Abstract summary: 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.
• Score: 45.87981728307819
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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