What does this research mean for the field?

The sequence of marginals obtained from the iterations of the Sinkhorn algorithm converges to an absolutely continuous curve on the 2-Wasserstein space, which can be characterized as a Wasserstein mirror gradient flow. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish the continuous-time limit of the marginal distributions generated by the Sinkhorn algorithm as entropy regularization approaches zero.

August 30, 2026The Annals of ProbabilityOpen Access

Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm

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Authors

NDNabarun DebWoodlawn School YKYoung‐Heon KimUniversity of British Columbia SPSoumik PalIndian Institute of Technology Guwahati

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Implication

Theoretical analysis demonstrates that Sinkhorn algorithm marginals converge to continuous gradient flows in optimal transport, indicating deep links to diffusion equations.

Key Points

To establish the continuous-time limit of the marginal distributions generated by the Sinkhorn algorithm as entropy regularization approaches zero.
Analyzed the scaling limits of Sinkhorn iterations and the iterative proportional fitting procedure on joint densities as regularization parameter ε vanishes and iterations scale as 1/ε.
Formulated a Wasserstein mirror gradient flow framework using relative entropy gradients and squared Wasserstein distance functionals.
Linked the resulting continuous flow to parabolic Monge–Ampère partial differential equations and McKean–Vlasov stochastic diffusions.
Proved that the sequence of marginal distributions converges to an absolutely continuous curve on 2-Wasserstein space termed the Sinkhorn flow.
Established conditions for exponential convergence of the limiting flow and showed its velocity field corresponds to the metric derivative in linearized optimal transport.
Constructed an explicit McKean–Vlasov diffusion process whose marginal probability distributions precisely trace the Sinkhorn flow.

Cite This Study

Deb et al. (2026) studied this question.

synapsesocial.com/papers/6a93efc06c1a8fb52e79bc8d https://doi.org/10.1214/25-aop1801

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Implication

Theoretical analysis demonstrates that Sinkhorn algorithm marginals converge to continuous gradient flows in optimal transport, indicating deep links to diffusion equations.

Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm

Discussion

Member takes

Implication

Key Points

Cite This Study

Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm

Discussion

Member takes

Implication

Key Points

Cite This Study

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