Optimal Transport Regularization (OTReg)

• OTReg is a framework that uses optimal transport either as a regularizer within inverse problems or as a constraint on the transport plan to preserve geometric structure.
• It unifies diverse methods in image restoration, speech-text alignment, and graph optimization by balancing smoothness, sparsity, and robustness.
• Algorithmic implementations range from Sinkhorn and mirror descent to Newton-type and interior-point methods, ensuring efficient and adaptable computation.

Optimal Transport Regularization (OTReg) denotes a family of constructions in which optimal transport is either used as a regularizer inside a larger variational model or is itself regularized by additional convex, nonconvex, or structural terms. In the cited literature, this label covers, among other uses, the dual-Lipschitz or Wasserstein-1 penalty for image restoration, entropic and non-entropic regularizations of Kantorovich transport, adaptive and stochastic regularization schedules in semi-discrete OT, and transport-based alignment losses for representation learning. Across these formulations, the common role of OTReg is to preserve geometric structure while modifying optimization, statistical, or robustness properties in a controlled way (Huang et al., 19 Mar 2025, Dessein et al., 2016).

1. Scope and canonical formulations

In the cited literature, OTReg is used for several mathematically distinct constructions. One recurrent template regularizes the transport plan itself: $$W_R(\mu,\nu)\;=\;\min_{\pi\in\Pi(\mu,\nu)}\;\Bigl\{\langle \pi,c_0\rangle+\epsilon\,R(\pi)\Bigr\},$$ where $R$ is a convex penalty on the coupling $\pi$. Another template uses an OT-derived quantity as a penalty inside a separate inverse problem or learning objective, as in image restoration or speech-text alignment. A third template constrains structure directly, for example by imposing per-row regularity budgets or explicit cardinality constraints on the plan (Paty et al., 2020, Assel et al., 2023, Liu et al., 2022).

Usage	Representative formulation	Role
OT as a regularizer in inverse problems	$$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$	cartoon-texture-noise decomposition
Regularized OT on couplings	$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$	smoothness, robustness, tractability
OT-based alignment loss	$\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$	speech-text alignment

The first line is exemplified by the image-restoration model

$$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$

which turns a Wasserstein-1 quantity into a regularizer on the image variable. The second line includes entropic, Orlicz-space, quadratic, Rényi, $\beta$-potential, and other plan regularizations. The third line appears in spoken LLMs, where an optimal transport plan between speech and transcript embeddings is computed and then converted into a differentiable regularization loss (Huang et al., 19 Mar 2025, Lorenz et al., 2019, Xu et al., 11 Aug 2025).

2. OTReg in image restoration and inverse imaging

In "Image Restoration Models with Optimal Transport and Total Variation Regularization" (Huang et al., 19 Mar 2025), OTReg is formulated from the Kantorovich–Rubinstein dual representation of the Wasserstein-1 distance. For nonnegative densities $\mu,\nu$ on a bounded domain $\Omega\subset\mathbb R^2$ with the same total mass,

$R$0

equivalently with $R$1. By convex duality,

$R$2

Setting $R$3, the OT regularizer becomes

$R$4

A central theoretical point is the relation to Meyer’s $R$5-norm. The paper states

$R$6

and identifies the dual-Lipschitz norm as a negative-Sobolev norm of order $R$7, namely the case $R$8 in

$R$9

with $\pi$0 recovering Meyer’s $\pi$1-norm. The combined model studied in practice is

$\pi$2

where $\pi$3. In the stated interpretation, $\pi$4 enforces piecewise constant cartoon structure, $\pi$5 retains fine oscillations, and the $\pi$6 term forces $\pi$7 (Huang et al., 19 Mar 2025).

The numerical scheme alternates between an OT subproblem in $\pi$8, solved by a Primal–Dual Hybrid Gradient method, and a TV subproblem in $\pi$9, solved by an Augmented Lagrangian Method. The OT step uses the saddle-point Lagrangian

$$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$0

with updates $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$1, $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$2, and $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$3. The TV step rewrites $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$4 and applies ALM or ADMM with a closed-form shrinkage update for $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$5. Replacing TV by the nonconvex "Log-TV" of Zhu et al. yields "KR–Log-TV," whose reported effect is to reduce staircasing and keep sharper edges. On "House," "Butterfly," "Parrot," "Barbara," and related tests with additive white Gaussian noise $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$6, the reported observations are that KR–TV preserves more meaningful contrast than ROF, KR–Log-TV further suppresses staircasing, and PSNR improved $$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$7–$$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$8 over ROF in the reported tests (Huang et al., 19 Mar 2025).

A related inverse-imaging formulation appears in "Parallel Unbalanced Optimal Transport Regularization for Large Scale Imaging Problems" (Lee et al., 2019). There the unbalanced OT regularizer is written in Beckmann form as

$$\displaystyle TV(u)+\frac{\alpha}{2}\|f-u-v\|_2^2+\lambda\|v\|_{Lip*}$$9

The stated advantages are linear optimization-variable complexity, a fully parallelizable proximal solver with $$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$0 per iteration, and superior empirical performance on dynamical tracking applications in synthetic and real video (Lee et al., 2019).

3. Regularizing the transport problem itself

A large part of the OTReg literature concerns penalties or constraints placed directly on the transport plan. In "Regularized Optimal Transport and the Rot Mover's Distance" (Dessein et al., 2016), the primal regularized problem is

$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$1

and is shown to be equivalent to a matrix-nearness problem with respect to a Bregman divergence: $$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$2 This yields the "rot mover’s distance" and a unified algorithmic framework encompassing Sinkhorn-type and non-Sinkhorn-type regularizers (Dessein et al., 2016).

Other works broaden the class of admissible penalties. "Orlicz-space regularization for optimal transport and algorithms for quadratic regularization" (Lorenz et al., 2019) replaces Radon plans by densities $$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$3 and solves

$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$4

The paper derives the predual

$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$5

and states strong duality and existence of primal solutions. For $$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$6 regularization with $$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$7, it also gives cyclic projection, dual gradient descent, simple fixed point, and Nesterov-accelerated methods (Lorenz et al., 2019).

Quadratic regularization is emphasized in graph settings. "Quadratically-Regularized Optimal Transport on Graphs" (Essid et al., 2017) studies

$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$8

states that quadratic regularization preserves sparsity and ensures a unique solution, and derives the dual

$$\displaystyle \min_{\pi\in\Pi(\mu,\nu)} \langle \pi,c_0\rangle+\epsilon R(\pi)$$9

whose Hessian is a weighted graph Laplacian. A damped Newton method then reduces each step to a Laplacian linear solve (Essid et al., 2017). "A regularized Interior Point Method for sparse Optimal Transport on Graphs" (Cipolla et al., 2023) introduces coupled primal–dual proximal regularization,

$\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$0

adds a log-barrier, sparsifies the shifted Laplacian normal equations, and proves inner-IPM complexity $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$1 (Cipolla et al., 2023).

Robustness-motivated OTReg replaces entropy by alternative divergences. "Robust computation of optimal transport by $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$2-potential regularization" (Nakamura et al., 2022) uses

$\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$3

and solves

$\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$4

The paper states that, under a cost-separation condition on outliers, the intermediate solution satisfies $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$5 for all outlier columns $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$6, so no mass is transported onto true outliers (Nakamura et al., 2022). "Interpolating between Optimal Transport and KL regularized Optimal Transport using Rényi Divergences" (Bresch et al., 2024) proposes

$\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$7

with $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$8. Its interpolation theorem states that, for fixed $\displaystyle L_{total}=L_{CE}+\lambda_{OT}L_{OT}$9, $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$0 recovers KL-regularized OT and $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$1 recovers unregularized OT, while for fixed $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$2, $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$3 also recovers unregularized OT (Bresch et al., 2024).

Structural regularization can be imposed more explicitly. "Sparsity-Constrained Optimal Transport" (Liu et al., 2022) solves

$$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$4

where $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$5. The paper states that $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$6 recovers unregularized OT plus a constant shift, while large $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$7 recovers $$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$8-regularized OT, thus making the method a middle ground between the two (Liu et al., 2022). "Optimal Transport with Adaptive Regularisation" (Assel et al., 2023) replaces a global budget by per-row or per-column budgets. In the row-wise version,

$$R_{OT}(u)=\|u-u_0\|_{Lip*} = \inf_m\left\{\int_\Omega |m|\,dx:\operatorname{div}m=u-u_0\right\},$$9

with

$\beta$0

Its stated purpose is to enforce a minimum of smoothing per point, in contrast to a single global regularization parameter (Assel et al., 2023).

A final conceptual reformulation is given in "Regularized Optimal Transport is Ground Cost Adversarial" (Paty et al., 2020). There any convex regularization is shown, via Fenchel duality, to admit an adversarial ground-cost interpretation: $\beta$1 For entropy, the corresponding adversarial cost satisfies

$\beta$2

This places OTReg not only in a computational but also in a robust-optimization framework (Paty et al., 2020).

4. Semi-discrete OTReg and adaptive regularization schedules

Semi-discrete OT provides a distinct OTReg regime in which the source measure is continuous and the target is discrete. In "Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularization" (2405.14459), the entropic semi-dual objective is

$\beta$3

with

$\beta$4

The stochastic gradient is

$\beta$5

and the proposed DRAG algorithm uses a decreasing schedule $\beta$6, projected SGD, and Polyak averaging. The paper states that this yields nearly minimax rates, including $\beta$7 mean-square error to the true unregularized potential and $\beta$8 error for the OT map, with lower bounds $\beta$9 for the potential and $\mu,\nu$0 for the map (2405.14459).

"Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport" (Genans et al., 31 Oct 2025) sharpens this perspective. With $\mu,\nu$1, $\mu,\nu$2, a projection onto a bounded convex set $\mu,\nu$3, and Polyak averaging, DRAG is stated to achieve an unbiased $\mu,\nu$4 sample and iteration complexity for both the OT cost and the potential estimation, and a $\mu,\nu$5 rate for the OT map. The stated explanation is that early iterations use large $\mu,\nu$6 for strong convexity and low variance, while later iterations use small $\mu,\nu$7 so that the entropic bias vanishes (Genans et al., 31 Oct 2025).

A complementary route appears in "Characterizing and computing solutions to regularized semi-discrete optimal transport via an ordinary differential equation" (Nenna et al., 3 Apr 2025). Setting $\mu,\nu$8, the dual potential $\mu,\nu$9 is characterized by

$\Omega\subset\mathbb R^2$0

with explicit initial condition at $\Omega\subset\mathbb R^2$1. The paper states global strong convexity of $\Omega\subset\mathbb R^2$2 on the affine subspace $\Omega\subset\mathbb R^2$3, invertibility of the Hessian on that subspace, and a third-order Runge–Kutta discretization. Its reported numerical finding is that the ODE method is competitive with Newton’s method for squared Euclidean cost and more robust for higher powers of the Euclidean distance (Nenna et al., 3 Apr 2025).

These semi-discrete developments build on the earlier semi-dual regularization framework of "Semi-dual Regularized Optimal Transport" (Cuturi et al., 2018), which shows that entropic regularization replaces nonsmooth $\Omega\subset\mathbb R^2$4-transforms by soft $\Omega\subset\mathbb R^2$5-transforms and turns several OT-based variational problems, including barycenters and gradient flows, into smooth convex problems with closed-form gradients and Hessians (Cuturi et al., 2018).

5. Algorithms and computational regimes

Algorithmically, OTReg spans matrix scaling, Bregman projection, primal–dual splitting, Newton-type methods, mirror descent, and interior-point methods. For entropic regularization, the baseline is Sinkhorn’s algorithm, which alternates diagonal rescalings of the Gibbs kernel $\Omega\subset\mathbb R^2$6. "Numerical Methods for Large-Scale Optimal Transport" (Tupitsa et al., 2022) states $\Omega\subset\mathbb R^2$7 cost per Sinkhorn iteration and emphasizes that entropy yields a strongly convex surrogate and smooth dual variables. It also presents accelerated primal–dual gradient methods that can use any strongly-convex regularization (Tupitsa et al., 2022).

Acceleration of Sinkhorn-type solvers is itself an OTReg topic. "Overrelaxed Sinkhorn-Knopp Algorithm for Regularized Optimal Transport" (1711.01851) introduces overrelaxed Bregman projections with parameter $\Omega\subset\mathbb R^2$8, proves global convergence using a Lyapunov function $\Omega\subset\mathbb R^2$9, and reports a gain in convergence speed by an order of magnitude in certain low-$R$00 regimes (1711.01851).

For general smooth convex regularizers, "Regularized Optimal Transport and the Rot Mover’s Distance" (Dessein et al., 2016) develops the alternate scaling algorithm and the non-negative alternate scaling algorithm, both based on Dykstra’s algorithm with alternate Bregman projections. In the separable case, row and column projections can be solved by Newton–Raphson updates, and a sparse extension is given for high-dimensional settings (Dessein et al., 2016). "Interpolating between Optimal Transport and KL regularized Optimal Transport using Rényi Divergences" (Bresch et al., 2024) instead uses a nested mirror-descent scheme,

$R$01

combining a mirror step with a Sinkhorn projection onto the transport polytope (Bresch et al., 2024).

Graph-structured problems admit more specialized second-order methods. In quadratically-regularized graph OT, the dual Hessian is a weighted graph Laplacian, so a damped Newton method requires only Laplacian linear solves (Essid et al., 2017). In sparse graph OT with primal–dual proximal stabilization, the normal equations involve the shifted Laplacian-plus-mass matrix

$R$02

and $R$03-sparsification drops inactive columns of $R$04 while preserving full rank through the $R$05 term (Cipolla et al., 2023).

Outside pure transport problems, algorithm design follows the induced variational structure. The image-restoration KR–TV model alternates PDHG for the OT subproblem and ALM or ADMM for the TV subproblem, with FFT-based linear solves when the imaging operator is convolutional (Huang et al., 19 Mar 2025). The speech alignment OTReg computes an entropic OT plan by Sinkhorn, with each Sinkhorn iteration costing $R$06, and reports that with $R$07–$R$08 Sinkhorn steps the overhead is manageable (Xu et al., 11 Aug 2025).

6. Applications and empirical behavior

Image restoration is a canonical OT-as-regularizer application. For KR–TV and KR–Log-TV, the reported empirical pattern is contrast preservation, residuals that keep mostly true noise rather than structure, suppression of staircasing, and sharper edges than ROF or pure Log-TV. The reported PSNR gain is $R$09–$R$10 over ROF on the stated denoising tests (Huang et al., 19 Mar 2025).

In spoken LLMs, "Optimal Transport Regularization for Speech Text Alignment in Spoken LLMs" (Xu et al., 11 Aug 2025) formulates speech-text alignment as entropic OT between transformed speech embeddings $R$11 and unique transcript embeddings $R$12, with cost

$R$13

The OTReg loss is

$R$14

and the total objective is $R$15. In the reported English ASR results, Base + OTReg with $R$16 gives $R$17 WER on CoVoST-2 and $R$18 on FLEURS, compared with $R$19 and $R$20 for the CE-only base model; the paper states that OTReg enhances speech-text alignment, mitigates the modality gap, and improves cross-domain generalization (Xu et al., 11 Aug 2025).

Regularity-based OTReg has also been used for map estimation and domain adaptation. "Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport" (Paty et al., 2019) imposes $R$21, obtains a transport map with two-sided distortion bounds, and reports that the SSNB estimator attains much smaller error constants in high dimension, domain-adaptation accuracy comparable to the best regularized OT methods with mean accuracy $R$22–$R$23, and smooth, contrast-controlled color mappings in a color-transfer task (Paty et al., 2019). "Recovery Bounds on Class-Based Optimal Transport: A Sum-of-Norms Regularization Framework" (Rahbar et al., 2019) adds row and column sum-of-norms penalties to encourage block-diagonal couplings and reports exact class-structure recovery under the stated geometric assumptions; on an MNIST→USPS imbalanced-class task, the best OT-SON score in the excerpt is $R$24, compared with $R$25 for OT-l1l2 and $R$26 for Sinkhorn (Rahbar et al., 2019). "Optimal Transport with Adaptive Regularisation" (Assel et al., 2023) reports that EOTARI-d reaches $R$27 on MNIST→USPS versus $R$28 for EOT, while QOTARI-t reaches $R$29 versus $R$30 for QOT (Assel et al., 2023).

Robustness-oriented OTReg has been evaluated on contaminated data. For $R$31-potential regularization, the reported toy example shows that entropic-OT jumps from approximately $R$32 to approximately $R$33 under a few outliers, whereas $R$34-OT stays at approximately $R$35, and outlier detection via zero column mass achieves $R$36 accuracy on MNIST vs Fashion-MNIST contamination (Nakamura et al., 2022). On sparse graphs, the proximal-stabilized regularized IPM is reported to outperform Lemon once $R$37, with experiments on graphs up to $R$38M edges and empirical scaling $R$39 versus Lemon’s $R$40 (Cipolla et al., 2023). In large-scale imaging, unbalanced Beckmann OTReg is reported to improve support recovery and rMSE in dynamical tracking, and RPCA+UOT-DF is reported to recover a compressed real video foreground where RPCA, RPCA+L1-DF, and RPCA+BOT-DF fail (Lee et al., 2019).

7. Interpretive issues, recurring trade-offs, and misconceptions

A persistent misconception is that OTReg is a single method. The cited literature instead uses the term for several non-equivalent operations: penalizing couplings, constraining couplings, using transport-based norms as regularizers in inverse problems, imposing regularity on Brenier potentials, and building alignment losses from optimal plans. This plurality is visible already from the contrast between KR–TV image restoration, entropic and Orlicz-regularized Kantorovich problems, adaptive regularization on rows and columns, and OT-based losses in spoken LLMs (Huang et al., 19 Mar 2025, Lorenz et al., 2019, Xu et al., 11 Aug 2025).

A second misconception is that regularization is only a computational device. The cited works assign it broader roles. In the Brenier-potential framework, regularity is itself the regularizer, with smoothness and strong convexity used to mitigate the curse of dimensionality and to enable out-of-sample evaluation (Paty et al., 2019). In class-based OT, a sum-of-norms penalty is used to recover block structure under strong cyclical monotonicity and related geometric assumptions (Rahbar et al., 2019). In the ground-cost adversarial formulation, any convex plan regularization can be reinterpreted as an adversarial perturbation of the ground cost (Paty et al., 2020).

A third recurrent issue is the bias–tractability trade-off. Entropic regularization yields smooth objectives and efficient Sinkhorn solvers, but introduces bias and fully dense plans. Quadratic regularization preserves sparsity on graphs; $R$41-potential regularization is designed for robustness to outliers; explicit cardinality constraints give direct control over the number of nonzeros; and adaptive or decreasing regularization schedules are introduced precisely to reduce or remove entropic bias while retaining computational advantages (Essid et al., 2017, Nakamura et al., 2022, Liu et al., 2022, Genans et al., 31 Oct 2025). This suggests that OTReg is best understood not as a single regularizer, but as a design space for trading off geometry preservation, sparsity, smoothness, robustness, and solver structure.