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arxiv: 2307.16421 · v2 · submitted 2023-07-31 · 🧮 math.PR · math.AP· stat.ML

Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm

Nabarun Deb , Young-Heon Kim , Soumik Pal , Geoffrey Schiebinger This is my paper

Pith reviewed 2026-05-24 07:54 UTC · model grok-4.3

classification 🧮 math.PR math.APstat.ML
keywords Sinkhorn algorithmWasserstein spacemirror gradient flowoptimal transportMonge-Ampère equationMcKean-Vlasov diffusioniterative proportional fittingrelative entropy
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The pith

The Sinkhorn algorithm converges in a scaled limit to a Wasserstein mirror gradient flow on the space of probability measures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that iterating the Sinkhorn algorithm on joint densities, with regularization parameter epsilon and number of iterations scaled as 1/epsilon, produces marginal distributions that converge to an absolutely continuous curve in the 2-Wasserstein space. This limiting curve is identified as a mirror gradient flow in Wasserstein space, with the mirror map given by half the squared Wasserstein distance to one marginal and the gradient taken from the relative entropy with respect to the other. A reader would care because this bridges a widely used computational method in optimal transport to a continuous-time dynamical system, opening the door to PDE-based analysis of its behavior and long-term properties. The work also links the flow to the parabolic Monge-Ampère equation and constructs an associated McKean-Vlasov process.

Core claim

The sequence of marginals from the Sinkhorn iterations converges to an absolutely continuous curve on the 2-Wasserstein space as ε → 0 with iterations scaled as 1/ε. This Sinkhorn flow is a Wasserstein mirror gradient flow, where the gradient is that of the relative entropy functional with respect to one marginal and the mirror is half of the squared Wasserstein distance functional from the other marginal. The norm of its velocity field can be interpreted as the metric derivative with respect to the linearized optimal transport distance. An equivalent description is the parabolic Monge-Ampère PDE. Conditions for exponential convergence are derived, and a McKean-Vlasov diffusion isconstructed

What carries the argument

The Sinkhorn flow, the limit of scaled Sinkhorn iterations, acting as a Wasserstein mirror gradient flow with relative entropy gradient and Wasserstein distance mirror.

If this is right

  • The limiting flow satisfies the parabolic Monge-Ampère PDE.
  • Exponential convergence of the flow to equilibrium holds under the derived conditions.
  • Marginals of a constructed McKean-Vlasov diffusion coincide with the Sinkhorn flow.
  • The speed of the flow equals the metric derivative with respect to the linearized optimal transport distance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Sinkhorn iterations could serve as a practical numerical scheme to approximate solutions of the parabolic Monge-Ampère equation.
  • The mirror flow structure may extend to other regularized optimal transport algorithms beyond the classical Sinkhorn procedure.
  • The McKean-Vlasov construction opens a route to particle-based simulations of the flow for sampling or mean-field models.

Load-bearing premise

The number of Sinkhorn iterations must scale precisely as one over the regularization parameter epsilon, together with suitable technical conditions on the joint densities.

What would settle it

Numerical computation showing that marginals from Sinkhorn iterations with N exactly 1/epsilon fail to approach the solution of the parabolic Monge-Ampère PDE in the 2-Wasserstein metric as epsilon goes to zero, for some initial joint densities meeting the paper's assumptions.

Figures

Figures reproduced from arXiv: 2307.16421 by Geoffrey Schiebinger, Nabarun Deb, Soumik Pal, Young-Heon Kim.

Figure 1
Figure 1. Figure 1: Convergence of the Sinkhorn iterates (ρ ε k , k ∈ N) to an absolutely continuous curve (the Sinkhorn flow) in the 2- Wasserstein space, with k = O(1/ε), as ε → 0. Here, for a given ε > 0, we view the points on the corresponding curves as a dis￾cretization of an absolutely continuous curve with O(1/ε) points. denotes the standard Kullback-Leibler (KL) divergence and ε > 0 is called the regularization parame… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the Sinkhorn flow (ρt, t ≥ 0). On the left, ∇ut is the Brenier map transporting ρt to e −g . (ut, t ≥ 0) satisfies the parabolic Monge-Amp`ere PDE: ∂ ∂t∇ut = ∇WKL(ρt ∥ e −f ). See Theorem 1.3. On the right, wt = u ∗ t is the corresponding family of convex conjugates. ∇wt = (∇ut) −1 transports e −g to ρt. For s < t, t − s ≈ 0, the difference ∇wt − ∇ws is approximately vs ◦ ∇ws, where vs is the … view at source ↗
read the original abstract

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to zero and the number of iterations is scaled as $1/\varepsilon$ (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. An equivalent description of this flow is provided by the parabolic Monge-Amp\`{e}re PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a convergence result for the Sinkhorn algorithm (or IPFP) applied to joint densities: as the regularization parameter ε tends to zero and the number of iterations is scaled proportionally to 1/ε (under additional technical assumptions on the densities), the sequence of marginals converges to an absolutely continuous curve in the 2-Wasserstein space. This limiting curve, termed the Sinkhorn flow, is identified as a Wasserstein mirror gradient flow, where the gradient is that of the relative entropy with respect to one marginal and the mirror is half the squared Wasserstein distance to the other marginal. The flow is also shown to satisfy a parabolic Monge-Ampère PDE, conditions for exponential convergence are derived, and a McKean-Vlasov diffusion is constructed whose marginals follow the flow.

Significance. If the main convergence theorem holds, the paper makes a significant contribution by bridging discrete iterative algorithms in optimal transport with continuous gradient flow dynamics in Wasserstein space through the new notion of Wasserstein mirror gradient flows. The explicit connection to the parabolic Monge-Ampère equation and the construction of the associated diffusion process provide concrete analytical tools. The result is self-contained with stated assumptions and offers potential for further developments in mean-field limits and numerical analysis of OT methods.

minor comments (3)
  1. [Abstract] Abstract: the scaling 'number of iterations is scaled as 1/ε' should be stated more precisely (e.g., whether N_ε ∼ C/ε for a fixed C or an exact floor function) since this scaling is load-bearing for the convergence statement.
  2. [References] The citation to Berman (2020) appears in the abstract and introduction but lacks a corresponding entry in the bibliography; this should be added for completeness.
  3. [§2] §2: the notation distinguishing the regularized joint densities from their marginals is introduced inline but would benefit from a short dedicated notation subsection to improve readability for readers unfamiliar with IPFP variants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the main results, and the recommendation for minor revision. The report correctly identifies the core contribution as the convergence of Sinkhorn iterations (with appropriate scaling) to a Wasserstein mirror gradient flow, along with the connections to the parabolic Monge-Ampère equation and the associated McKean-Vlasov diffusion. We are pleased that the significance of introducing this new notion of mirror gradient flows in Wasserstein space is recognized.

Circularity Check

0 steps flagged

No significant circularity; self-contained limit theorem

full rationale

The paper establishes a convergence result: under the scaling of iterations ~1/ε and technical assumptions on joint densities, the Sinkhorn/IPFP marginals converge in 2-Wasserstein space to an absolutely continuous curve (the Sinkhorn flow). This limit is then identified as a Wasserstein mirror gradient flow (with relative entropy gradient and W2 mirror) whose velocity norm matches the LOT metric derivative, and which satisfies a parabolic Monge-Ampère PDE previously linked to Sinkhorn by Berman (2020). The central derivation is a limit theorem with assumptions stated upfront; the new concept is introduced by direct analogy to Euclidean mirror flows rather than by self-referential definition or fitted parameters. No load-bearing self-citations, ansatz smuggling, or reductions by construction appear in the argument structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on unspecified technical assumptions for the densities and scaling; no free parameters, invented entities, or explicit axioms are visible in the abstract.

axioms (1)
  • domain assumption technical assumptions on joint densities and iteration scaling as ε → 0
    Abstract states these are required for the sequence to converge to an absolutely continuous curve.

pith-pipeline@v0.9.0 · 5765 in / 1185 out tokens · 37308 ms · 2026-05-24T07:54:22.320492+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The entropic optimal (self-)transport problem: Limit distributions for decreasing regularization with application to score function estimation

    math.ST 2024-12 unverdicted novelty 7.0

    Provides asymptotic distributions for entropic OT plans and potentials under vanishing regularization and links self-transport barycentric projections to score functions.

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