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Impact of update ordering on SeqOT Sinkhorn guarantees

Determine whether the convergence guarantees established for the Sinkhorn iteration for sequentially composed optimal transport—where the boundary dual variables f_i for i=2,…,M are updated first and the edge variables f_1 and f_{M+1} are updated afterwards—remain valid when the order of these variable updates is changed (for example, updating edge variables before boundary variables or using other permutations).

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Background

The paper introduces a Sinkhorn-type algorithm for sequentially composed optimal transport (SeqOT) derived from the dual formulation. A key design choice is the specific ordering of updates: first updating the variables associated with the internal boundaries between sequential transport plans, and then updating the variables associated with the edge marginals.

The authors emphasize that many proofs in the paper—such as those establishing convergence in the Hilbert (pseudo-)metric and complexity results for the M=2 case—explicitly depend on this particular update ordering. Whether similar theoretical results persist under alternative update orders is not established and is highlighted as unclear.

References

We carefully choose this ordering of the variable updates, that is, we first update the variables on the boundaries, and then update the variable on the edges. In fact, most proofs in this paper explicitly depend on this ordering, and it is unclear whether we can get similar results by changing the ordering.

— Sinkhorn Algorithm for Sequentially Composed Optimal Transports (2412.03120 - Watanabe et al., 4 Dec 2024) in Remark, Section 2 (Problem Formulation and the Sinkhorn Algorithm)