Sinkhorn Divergence Drift Dynamics

• Sinkhorn divergence drift is defined as the first-order motion induced by a debiased entropic optimal transport functional, combining attractive transport and self-correction.
• It characterizes a Wasserstein gradient flow where target attraction is balanced by a self-interaction term that removes entropic shrinkage bias.
• The framework underpins both sample-based estimation and explicit Gaussian models, with applications in generative modeling and numerical optimization.

Searching arXiv for recent and foundational papers on Sinkhorn divergence drift and related gradient-flow formulations. arXiv search: "Sinkhorn divergence gradient flow Sinkhorn divergence drift generative". Sinkhorn divergence drift denotes the first-order motion induced by a Sinkhorn-type discrepancy between probability measures. In its most direct formulation, it is the Wasserstein gradient-flow velocity field associated with the debiased entropic optimal transport functional $S_\varepsilon(\rho,\nu)$, so that an evolving law $\rho_t$ moves toward a target $\nu$ by a cross-minus-self field built from entropic OT potentials (Zhu et al., 2024). Closely related work uses the same idea in nearby but non-identical senses: as an explicit affine drift for Gaussian source and target measures (Hardion et al., 11 Feb 2026), as a nonlocal potential flow in the geometry generated by Sinkhorn divergences themselves (Hardion et al., 18 Nov 2025), and as a particlewise barycentric drift for generative modeling (He et al., 12 Mar 2026). Across these settings, the characteristic structure is an attractive transport term toward the target together with a self-interaction correction that removes entropic shrinkage bias.

1. Terminological scope and basic object

The standard Sinkhorn divergence is the debiased entropic OT quantity

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$

where

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$

In the NSGF formulation, the “Sinkhorn-divergence drift” is precisely the time-dependent velocity field driving the Wasserstein gradient flow of $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$ (Zhu et al., 2024).

The terminology is not fully uniform. In generative drifting, a Sinkhorn drift is introduced through the generic template $V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$ with $F[q]=S_\varepsilon(q,p)$, and the resulting particle field is expressed through barycentric projections of entropic couplings rather than through dual potentials (Turan et al., 10 Mar 2026). By contrast, one computational paper uses “Sinkhorn divergence” for the regularized OT cost induced by the entropically regularized optimal plan, not for the debiased quantity above; in that notation,

$$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$

which is explicitly an upper bound on $W_p({\bf f},{\bf g})$ (Motamed, 2020). Any discussion of Sinkhorn divergence drift therefore depends on which definition of the underlying functional is being used.

A persistent structural feature nonetheless survives these differences. Whether the drift is written through Schrödinger potentials, barycentric projections, or metric tensors, it splits into a cross term involving the target and a self term involving the current measure. That cross-minus-self form is the operational signature of Sinkhorn debiasing.

2. Variational formulation and exact drift field

For a source distribution $\rho_t$0, target distribution $\rho_t$1, and evolving law $\rho_t$2 on $\rho_t$3, NSGF formulates the dynamics simultaneously as a Wasserstein gradient flow, a continuity equation,

$\rho_t$4

and a particle ODE,

$\rho_t$5

with energy $\rho_t$6 (Zhu et al., 2024).

The first variation is represented through Sinkhorn dual potentials. If $\rho_t$7 denotes the optimal dual potential for $\rho_t$8 and $\rho_t$9 the self-coupling potential for $\nu$0, then

$\nu$1

and the exact drift is

$\nu$2

At time $\nu$3,

$\nu$4

Because the field is a gradient field, the motion may also be written with the time-dependent potential $\nu$5, so that $\nu$6 (Zhu et al., 2024).

This representation makes the debiasing mechanism explicit. The term $\nu$7 is the attractive component pulling mass toward target regions favored by entropic OT, while $\nu$8 is the self-interaction correction that cancels the artificial spreading or attraction induced purely by regularization on the current law. The exact drift is therefore smoother than the $\nu$9-gradient of unregularized OT, but unlike raw entropic OT it is corrected by construction for entropic shrinkage bias (Zhu et al., 2024).

Sign conventions vary across the literature. In the Gaussian analysis the Wasserstein gradient flow is written as

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$0

so the velocity field is

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$1

which differs from the NSGF sign convention only by the continuity-equation convention used to encode steepest descent (Hardion et al., 11 Feb 2026).

3. Empirical estimation and generative realizations

A central feature of NSGF is that the Sinkhorn-divergence drift can be estimated from samples only. Given particles $$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$2 and target samples $$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$3, define empirical measures

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$4

From the associated empirical dual potentials $$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$5 and $$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$6, the empirical drift is

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$7

Training then uses velocity field matching: $$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$8 with a neural ODE

$$S_\varepsilon(\alpha,\beta) = \mathrm{OT}_\varepsilon(\alpha,\beta) -\frac12\mathrm{OT}_\varepsilon(\alpha,\alpha) -\frac12\mathrm{OT}_\varepsilon(\beta,\beta),$$9

The theoretical statement is mean-field consistency: as sample size grows, the empirical approximation converges to the true underlying velocity field (Zhu et al., 2024).

The same paper introduces NSGF++, a two-phase construction motivated by the behavior of the drift. The first phase follows the Sinkhorn flow to approach the image manifold quickly, reported as achievable in $$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$0 NFEs; the second phase refines samples along a simple straight flow (Zhu et al., 2024). This decomposes the role of Sinkhorn drift into a manifold-approaching transport component and a cheaper local refinement component.

Generative drifting gives a parallel particle-level interpretation. Standard drifting defines

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$1

with attractive and repulsive terms built from one-sided normalized Gibbs kernels. Sinkhorn drifting replaces those one-sided normalizations by entropic OT couplings obtained through two-sided Sinkhorn scaling, and for empirical particles yields

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$2

where $$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$3 and $$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$4 are barycentric projections of the cross and self entropic couplings (Turan et al., 10 Mar 2026). In that framework, $$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$5 Sinkhorn iteration recovers standard drifting, while $$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$6 gives the exact particlewise gradient-flow field of the Sinkhorn divergence (He et al., 12 Mar 2026).

This reinterpretation addresses an identifiability issue in prior drifting formulations. Because Sinkhorn divergence is definite, the continuous smooth-setting result is that

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$7

whereas the earlier one-sided drift did not have a comparably clean implication (He et al., 12 Mar 2026).

4. Explicit Gaussian Sinkhorn-divergence drift

When both source and target are Gaussian, the Sinkhorn-divergence drift becomes completely explicit. For

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$8

the Sinkhorn divergence is

$$\mathrm{OT}_\varepsilon(\alpha,\beta) = \min_{\pi\in \Pi(\alpha,\beta)} \int c(x,y)\,d\pi(x,y) + \varepsilon\,\mathrm{KL}(\pi\|\alpha\otimes\beta).$$9

with a closed covariance functional $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$0 built from matrix square roots and log-determinants (Hardion et al., 11 Feb 2026).

In this setting the Schrödinger potentials are quadratic, and the debiased gradient field is affine: $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$1 where $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$2 is an explicit matrix obtained from $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$3, $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$4, and $\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$5. Consequently, if

$\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$6

then the particle drift field is

$\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$7

and the infinite-dimensional PDE reduces to

$\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$8

The mean therefore converges explicitly as

$\mathcal F(\rho)=S_\varepsilon(\rho,\nu)$9

(Hardion et al., 11 Feb 2026).

The Gaussian theory also clarifies support effects. If the initial covariance is invertible, the flow converges globally to the target. If it is singular, convergence may fail, and singularity is preserved for almost every time. In the commuting case, each eigenvalue evolves by a scalar ODE balancing an attractive target term against a repulsive self term; this yields exponential convergence when source and target share support, but only $V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$0 decay when the target is concentrated on a strict subspace of the source support (Hardion et al., 11 Feb 2026). The self-correction that debiases entropic OT thus also resists variance collapse, which is why lower-dimensional targets produce qualitatively slower Sinkhorn drift.

5. Sinkhorn geometry beyond the balanced Wasserstein flow

A distinct line of work replaces not just the energy but the geometry itself. Instead of the classical JKO step

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$1

the Sinkhorn-JKO scheme uses

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$2

with $V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$3. The resulting continuous-time Sinkhorn potential flow is the operator inclusion

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$4

not, in general, a local continuity equation of the form $V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$5 (Hardion et al., 18 Nov 2025).

The paper resolves this apparent nonlocality by a nonlinear RKHS embedding

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$6

under which the dynamics become

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$7

The drift is then interpreted not as a flat Hilbert gradient system but as a monotone inclusion combining a skew operator with a normal-cone term. The theory proves existence, uniqueness, non-expansiveness in the embedded RKHS variable, and energy dissipation

$V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$8

together with convergence of $V(x)=-\nabla_x \frac{\delta F}{\delta q}(x)$9 to its minimal value. If $F[q]=S_\varepsilon(q,p)$0 has a unique minimizer $F[q]=S_\varepsilon(q,p)$1, then $F[q]=S_\varepsilon(q,p)$2 (Hardion et al., 18 Nov 2025).

The unbalanced theory extends the same logic to positive measures with varying total mass. The regularized cost is

$F[q]=S_\varepsilon(q,p)$3

and the corresponding unbalanced Sinkhorn divergence is

$F[q]=S_\varepsilon(q,p)$4

Its variational derivatives provide a drift on the cone of positive measures, with an additional mass-modulation component. In this setting the dynamics can favor transport, destruction, or creation of mass, whereas balanced OT only transports existing mass (Séjourné et al., 2019).

6. Discrete algorithmic drift, approximation, and numerical stability

At the level of entropic OT subproblems, Sinkhorn iteration itself admits a discrete drift interpretation. One mirror-descent analysis shows that the evolving marginal $F[q]=S_\varepsilon(q,p)$5 follows

$F[q]=S_\varepsilon(q,p)$6

equivalently

$F[q]=S_\varepsilon(q,p)$7

so standard Sinkhorn can be read as Bregman gradient descent for $F[q]=S_\varepsilon(q,p)$8 in a problem-specific KL geometry (2002.03758). This viewpoint concerns the entropic OT components from which Sinkhorn-divergence drifts are assembled, rather than the debiased combination itself.

For Sinkhorn barycenters, the first-order field is explicit. With fixed $F[q]=S_\varepsilon(q,p)$9,

$$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$0

the difference of a cross Sinkhorn potential and a self Sinkhorn potential. A Frank–Wolfe iteration then updates the current barycenter by

$$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$1

where $$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$2 minimizes the current linearized Sinkhorn-divergence field (Luise et al., 2019). In this sense the drift is realized as a sparse atom-adding evolution on measure space.

Approximation can change the drift in controlled but nontrivial ways. Positive-feature Sinkhorn replaces the dense Gibbs kernel by

$$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$3

so Sinkhorn iterations scale as $$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$4. The paper separates kernel approximation error from finite-iteration error and proves high-probability control of the regularized OT value; for Sinkhorn divergence, the induced value drift follows termwise by linearity. It also stresses that the approximation remains fully differentiable, although explicit bounds on gradient drift are not derived (Scetbon et al., 2020).

Backward differentiation through Sinkhorn layers introduces another source of drift. An implicit-differentiation framework derives exact KKT-based backward equations for entropic OT and emphasizes the distinction between two regimes: “optimal gradients for a suboptimal forward pass” versus “approximate gradients for an approximate forward pass.” The resulting error bounds show that gradient perturbations are controlled by $$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$5, with an explicit $$S_{p,\lambda}({\bf f},{\bf g})=\langle P_\lambda,C\rangle^{1/p},$$6 amplification in the cost-gradient bound. This identifies solver truncation, small regularization, and tiny transport entries as direct mechanisms for gradient instability in Sinkhorn-based objectives (Eisenberger et al., 2022).

Taken together, these results delimit the current understanding of Sinkhorn divergence drift. The field is mathematically explicit in some regimes, notably sample-based Wasserstein flows and Gaussian models; it admits principled geometric generalizations through Sinkhorn-JKO and unbalanced transport; and it can be approximated efficiently enough for large-scale learning. At the same time, terminology remains nonuniform, full continuum convergence outside special settings is not yet settled, and several approximation schemes provide value guarantees without a complete perturbation theory for the induced gradients (Hardion et al., 11 Feb 2026).