Dynamic Sinkhorn Recursion in OT

• Dynamic Sinkhorn Recursion is the continuous-time limit of the classical Sinkhorn algorithm, connecting discrete iterative methods with measure-valued flows in OT.
• It formulates the evolution via a parabolic Monge–Ampère system and interprets the updates as Wasserstein mirror gradient flows, ensuring convergence.
• The framework guarantees exponential convergence under log-Sobolev conditions and provides a stochastic interpretation through McKean–Vlasov diffusions.

Dynamic Sinkhorn Recursion refers to the continuous-time limit of the classical Sinkhorn or Iterative Proportional Fitting Procedure (IPFP) for entropy-regularized optimal transport (OT). In this framework, as the regularization parameter $\varepsilon \to 0$ and the number of iterations is scaled appropriately, alternative updates of the dual variables converge to a time-evolution in the $2$-Wasserstein space. This evolution, described by a parabolic Monge–Ampère PDE, is characterized as a Wasserstein mirror gradient flow for the relative entropy functional. The dynamic Sinkhorn recursion thus provides a bridge between discrete iterative OT algorithms and continuous-time, measure-valued flows, with associated exponential convergence results and a stochastic interpretation via McKean–Vlasov diffusions.

1. From Discrete Sinkhorn to Continuous-Time Evolution

The classical Sinkhorn algorithm for entropy-regularized OT solves for the optimal coupling between two densities $\mu(dx) = e^{-f(x)}dx$ and $\nu(dy) = e^{-g(y)}dy$ by iteratively updating dual “potential” functions:

• At iteration $k$, the $v$-potential is updated as

$$v_k(y) = \varepsilon \log \int \exp\left(\frac{x \cdot y - u_k(x) - f(x)}{\varepsilon}\right)\,dx$$

• The $u$-potential is updated via

$$u_{k+1}(x) = \varepsilon \log \int \exp\left(\frac{x \cdot y - v_k(y) - g(y)}{\varepsilon}\right)\,dy$$

At the level of the marginals, letting $\rho_k$ and $2$0 denote the $2$1- and $2$2-marginals of the entropic coupling at step $2$3,

$2$4

Each update enforces one marginal constraint via a multiplicative scaling.

Passing to the parabolic scaling limit, as $2$5 and the iteration index $2$6, the discrete increment $2$7 (similarly for $2$8) converges to a time derivative of a convex function $2$9. The resulting pair $\mu(dx) = e^{-f(x)}dx$0 solves a coupled system of PDEs known as the parabolic Monge–Ampère equations.

2. Parabolic Monge–Ampère System and Marginal Evolution

In the scaling limit, the dynamic Sinkhorn recursion is governed by

$\mu(dx) = e^{-f(x)}dx$1

where $\mu(dx) = e^{-f(x)}dx$2 is the negative log-density of the pushforward of $\mu(dx) = e^{-f(x)}dx$3 by $\mu(dx) = e^{-f(x)}dx$4. The $\mu(dx) = e^{-f(x)}dx$5-marginal at time $\mu(dx) = e^{-f(x)}dx$6 is thus $\mu(dx) = e^{-f(x)}dx$7, forming an absolutely continuous curve in $\mu(dx) = e^{-f(x)}dx$8, the $\mu(dx) = e^{-f(x)}dx$9-Wasserstein space.

For the density $\nu(dy) = e^{-g(y)}dy$0, the pair $\nu(dy) = e^{-g(y)}dy$1 satisfies the continuity equation

$\nu(dy) = e^{-g(y)}dy$2

with velocity field

$\nu(dy) = e^{-g(y)}dy$3

Alternatively, the time evolution of the Brenier map $\nu(dy) = e^{-g(y)}dy$4 is given by

$\nu(dy) = e^{-g(y)}dy$5

where $\nu(dy) = e^{-g(y)}dy$6 denotes the Wasserstein gradient.

3. Mirror Gradient Flows in Wasserstein Geometry

The dynamic Sinkhorn recursion is naturally interpreted as a mirror descent on the space of probability measures:

• The “potential” is $\nu(dy) = e^{-g(y)}dy$7
• The mirror is $\nu(dy) = e^{-g(y)}dy$8

The Wasserstein gradient of $\nu(dy) = e^{-g(y)}dy$9, $k$0, yields the mirror coordinate $k$1. The $k$2-mirror gradient of $k$3 is

$k$4

The resulting mirror-flow is described by

$k$5

which, in terms of the densities, leads back to the continuity equation with velocity field $k$6 above. The norm $k$7 coincides with the (infinitesimal) linearized OT (LOT) metric.

4. Exponential Convergence and Log-Sobolev Conditions

Suppose $k$8 satisfies a logarithmic Sobolev inequality with constant $k$9 and $v$0 has uniform spectral bounds so that $v$1. Then, the evolution enjoys a global exponential decay:

$v$2

This implies

$v$3

and, consequently,

$v$4

Exponential convergence is thus guaranteed under the joint log-Sobolev and Hessian conditions.

5. Sinkhorn Diffusion: McKean–Vlasov SDE Representation

The dynamic Sinkhorn recursion can be realized as the marginal law of a McKean–Vlasov diffusion:

$v$5

with

$v$6

Under reasonable regularity, $v$7 has marginal density $v$8 solving the Sinkhorn PDE.

In dual coordinates $v$9,

$$v_k(y) = \varepsilon \log \int \exp\left(\frac{x \cdot y - u_k(x) - f(x)}{\varepsilon}\right)\,dx$$0

where $$v_k(y) = \varepsilon \log \int \exp\left(\frac{x \cdot y - u_k(x) - f(x)}{\varepsilon}\right)\,dx$$1 is a stationary solution.

Thus, the dynamic Sinkhorn recursion has a precise stochastic interpretation as the distributional flow of a non-linear diffusion process.

6. Significance and Scope

Dynamic Sinkhorn recursion provides a continuum limit unifying finite-dimensional IPFP-type methods with the geometry of optimal transport and infinite-dimensional mirror descent. The formulation via parabolic Monge–Ampère PDEs places the evolution within the established theory of Wasserstein geometry, mirror flows, and stochastic processes on spaces of probability measures. The exponential convergence established under log-Sobolev assumptions links algorithmic performance with underlying functional inequalities and regularity of densities. The stochastic realization (Sinkhorn diffusion) further connects the recursion to mean-field limits and probabilistic interpretations, expanding its relevance in analysis, probability, and algorithmic optimal transport (Deb et al., 2023).