Quantitative stability in optimal transport for general power costs
Pith reviewed 2026-05-23 23:13 UTC · model grok-4.3
The pith
Optimal transport potentials and maps remain quantitatively stable under small perturbations of the target measure for power costs with exponent greater than 1.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For optimal transport problems with cost |x-y|^p where p is strictly larger than 1, if the source measure is log-concave and has bounded support, then the optimal potentials and the induced transport maps vary continuously with respect to perturbations of the target measure, and the size of the variation admits explicit bounds controlled by the distance between the two targets.
What carries the argument
Explicit quantitative stability estimates that bound the difference between optimal potentials (and maps) by a multiple of the perturbation size measured in the target measures.
If this is right
- The stability results apply uniformly to all Wasserstein distances with power exponent strictly larger than 1.
- The bounds are explicit and depend on the log-concavity and diameter of the source support.
- Both the potential functions and the transport maps inherit quantitative stability.
- The approach works with assumptions concentrated on the source measure, leaving the target largely unrestricted.
Where Pith is reading between the lines
- The explicit bounds could be applied to derive convergence rates when the target is replaced by its empirical version from samples.
- The technique might extend to stability under perturbations of the cost function rather than the measures.
- Analogous quantitative control could be sought when the source measure varies instead of the target.
Load-bearing premise
The source measure must be log-concave and have bounded support for the explicit stability bounds to hold.
What would settle it
A concrete counterexample with a log-concave bounded source measure, a cost with exponent p greater than 1, and a sequence of target measures approaching the original one whose optimal maps or potentials differ by more than the stated bound would disprove the claim.
read the original abstract
We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant for both theoretical and practical applications. Our results apply to a wide range of costs, including all Wasserstein distances with power cost exponent strictly larger than $1$ and leverage mostly assumptions on the source measure, such as log-concavity and bounded support. Our work provides a significant step forward in the understanding of stability of optimal transport problems, as previous results where mostly limited to the case of the quadratic cost.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. It provides explicit bounds on the stability of optimal transport potentials and maps for a wide range of costs, including all Wasserstein distances with power cost exponent p > 1. The results leverage assumptions primarily on the source measure, such as log-concavity and bounded support, extending previous work that was mostly limited to the quadratic cost.
Significance. If the explicit bounds hold under the stated assumptions, the work would advance stability theory in optimal transport by extending quantitative estimates beyond the quadratic case to general power costs p > 1. The emphasis on source-measure assumptions (log-concavity, bounded support) enables concrete, explicit bounds relevant for both theoretical analysis and applications involving Wasserstein distances.
minor comments (2)
- [Abstract] Abstract: the sentence 'previous results where mostly limited' contains a grammatical error ('where' should be 'were').
- [Abstract] Abstract: the claim of 'explicit bounds' is central but the abstract does not indicate the form of the bounds (e.g., dependence on p, dimension, or support radius); a brief indication would improve clarity for readers.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, the recognition of its novelty in extending quantitative stability results beyond the quadratic cost, and the recommendation for minor revision. The report contains no specific major comments requiring point-by-point responses.
Circularity Check
No significant circularity
full rationale
The paper derives explicit quantitative stability bounds for optimal transport potentials and maps under perturbations of the target measure, for costs including power costs with exponent p>1. The derivation relies on stated assumptions such as log-concavity and bounded support of the source measure, extending prior quadratic-cost results without any evident self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims to inputs by construction. The abstract and claims present the bounds as derived from these assumptions in a self-contained manner.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Source measure is log-concave
- domain assumption Source measure has bounded support
discussion (0)
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