Structured Optimal Transport

• Structured Optimal Transport is a family of optimal transport formulations that integrate structural priors to capture relational dependencies beyond fixed pairwise costs.
• It leverages nonlinear cost functions, submodular regularization, and structured couplings to yield transport plans that preserve group coherence and geometric relationships.
• The approach enables applications across graphs, cross-domain tasks, and latent variable models by fusing computational strategies with domain-specific structure.

Searching arXiv for recent and foundational papers on Structured Optimal Transport and closely related SOT variants. Structured Optimal Transport denotes a family of optimal-transport formulations that impose structure beyond a fixed pairwise ground cost. In the literature, the term has been used for a nonlinear generalization of discrete OT based on submodular costs over assignments, for feature transforms obtained by matching a set to itself, for cross-space models that learn a structured linear transform jointly with a transport plan, and for graph and multi-marginal constructions in which geometry, relations, or factorization constraints are built into the transport problem (Alvarez-Melis et al., 2017, Shalam et al., 2022, Sebbouh et al., 2023, Vayer et al., 2018, Haasler et al., 2020). At the same time, the acronym “SOT” is overloaded: it also denotes stochastic optimal transport, semimartingale optimal transport, simultaneous optimal transport, sliced optimal transport, and Sub-Optimal Transport in different parts of the literature (Mikami, 2023, Hou et al., 25 Jun 2026, Wang et al., 2022, Nguyen, 17 Aug 2025, Piombo et al., 4 Feb 2026). This suggests that “Structured Optimal Transport” is best understood as a research direction rather than a single canonical formalism.

1. Scope, nomenclature, and recurring design principles

A recurring theme is that classical OT treats each source–target assignment independently once the ground cost matrix is fixed, whereas structured variants let the cost, feasible couplings, or transport dynamics depend on additional relations. In the 2017 formulation, the explicit motivation is that OT’s ability to capture frequently occurring structure beyond the “ground metric” is limited (Alvarez-Melis et al., 2017). In later work, structure appears through feature–structure fusion on graphs, self-matching within a set, tree factorizations, learned sparse or low-rank transforms across spaces, or admissible semimartingale characteristics (Vayer et al., 2018, Shalam et al., 2022, Haasler et al., 2020, Sebbouh et al., 2023, Hou et al., 25 Jun 2026).

Formulation	Structural element	Representative paper
Discrete nonlinear SOT	Submodular cost on assignments	(Alvarez-Melis et al., 2017)
Self-Optimal-Transport	Instance-wise self-matching	(Shalam et al., 2022)
Cost-regularized OT across spaces	Structured linear transform $M$	(Sebbouh et al., 2023)
Fused Gromov-Wasserstein	Features + graph structure	(Vayer et al., 2018)
Tree-structured MMOT	Edge-factorized cost on a tree	(Haasler et al., 2020)
Semimartingale OT	Path laws with constrained characteristics	(Hou et al., 25 Jun 2026)

A common misconception is that Structured Optimal Transport refers to one fixed model. The literature does not support that reading. Another misconception is that “structure” is always introduced by entropic regularization alone. In several formulations, entropy is computationally important, but the actual structure comes from submodularity, tree factorization, vector-valued marginal constraints, anchor factorizations, sparse or low-rank transform penalties, or pathwise admissibility constraints (Alvarez-Melis et al., 2017, Haasler et al., 2020, Wang et al., 2022, Lin et al., 2020, Sebbouh et al., 2023).

2. Nonlinear costs and structured couplings

The foundational discrete formulation starts with measures

$$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$

the transportation polytope

$$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$

and classical OT

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$

Structured OT replaces the linear objective with the Lovász extension of a submodular set function $F$: $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$ where $B_F$ is the base polytope of $F$ (Alvarez-Melis et al., 2017).

This change makes the marginal cost of assigning one pair depend on what other assignments are chosen. The paper emphasizes cluster-wise concave costs, rank or volume costs, hierarchical structures, coverage functions, and graph or line-cut structures. In this view, classical OT is the modular special case: if $F$ is modular, then $B_F$ collapses to a single point and the problem reduces exactly to $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$0 (Alvarez-Melis et al., 2017).

The geometric effect is the emergence of structured couplings. Rather than arbitrary sparse or diffuse plans, the optimal $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$1 can become block-structured, coherent within groups, or neighborhood-preserving. The same paper interprets the minimax formulation as both a two-player zero-sum game and a robust-optimization problem in which the transport plan is optimized against a structured adversarial cost chosen from $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$2. Convexity follows from submodularity through the support-function identity $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$3, and under mild assumptions the induced distance $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$4 is a semi-metric (Alvarez-Melis et al., 2017).

Optimization exploits two projections: KL projection onto the transport polytope $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$5, solved by Sinkhorn-Knopp, and Euclidean projection onto $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$6. The paper proposes Mirror Descent with Averaging, saddle-point mirror descent, and saddle-point mirror-prox; the stated rates are $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$7 for MDA and $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$8 for mirror-prox (Alvarez-Melis et al., 2017). Empirically, this structure improves downstream transport-based learning: on MNIST $$\mu = \sum_{i=1}^n p_i^s\delta_{\mathbf{x}_i^s}, \qquad \nu = \sum_{j=1}^m p_j^t\delta_{\mathbf{x}_j^t},$$9 USPS and USPS $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$0 MNIST, Struct-OT achieved $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$1 and $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$2, compared with $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$3 and $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$4 for Group-Lasso OT, and syntax-aware Word Mover’s Distance improved from MSE $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$5, Spearman $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$6 to MSE $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$7, $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$8 on SICK (Alvarez-Melis et al., 2017).

3. Feature transforms, latent anchors, and structured assignment layers

A second major line of work uses OT to re-embed data or to force information flow through structured latent variables. The Self-Optimal-Transport feature transform takes a set of features $$M_{\mu,\nu} = \{ \gamma \in \mathbb{R}^{n \times m}_+ \mid \gamma \mathbf{1} = \mu,\ \gamma^T\mathbf{1} = \nu \},$$9, forms

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$0

and matches the set to itself by solving a fractional self-matching problem over the Birkhoff polytope. Its entropy-regularized form is

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$1

with $$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$2, $$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$3, to forbid self-matching (Shalam et al., 2022). The rows of $$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$4 are the transformed features. Distances between rows encode both direct original similarity and third party agreement regarding similarity to other features in the set. The transform is described as efficient, differentiable, equivariant, parameterless and probabilistically interpretable; it is also transductive and has adaptive dimensionality tied to set size. The reported applications include unsupervised clustering, few-shot classification, and person re-identification (Shalam et al., 2022).

A closely related idea is to force transport through a small number of latent anchor points. In Latent Optimal Transport, mass moves from source points to source anchors, from source anchors to target anchors, and from target anchors to target points, yielding a factored coupling

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$5

with $$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$6 (Lin et al., 2020). The paper frames this as joint clustering and alignment, and argues that the resulting plan is more robust to noise, outliers, and domain shift than direct sample-level OT.

Substructural Optimal Transport for cross-domain activity recognition imposes structure at an intermediate scale between classes and individual samples. It clusters source activities within each class via GMM and target data via GMM, represents substructures either by centers or Gaussians, computes source weights by a partial-OT step, and solves

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$7

with a group-sparse regularizer

$$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$8

to encourage class-consistent transport (Lu et al., 2021). On four public HAR datasets, the paper reports average accuracies of $$\min_{\gamma \in M_{\mu,\nu}} \langle \gamma, C \rangle.$$9 for $F$0 and $F$1 for $F$2, compared with $F$3 for the best non-SOT baseline STL (Lu et al., 2021).

Structured regularization can also target latent-to-data couplings in generative modeling. Informative OT-based GANs start from entropically regularized OT, interpret the optimal coupling as a proxy joint distribution between latent and data variables, and add a negative KL regularizer on the $F$4-marginal of that coupling: $F$5 By convex duality, this is equivalent to a perturbation of the ground cost $F$6, biasing the plan toward informative latent dimensions (Bréchet et al., 2019).

A recent example integrates entropic OT directly into a diffusion model’s attention mechanism. SceneTransporter formulates patch-to-part routing at denoising step $F$7 as

$F$8

then uses the resulting transport plan to gate cross-attention and impose exclusive, one-to-one routing of image patches to part-level 3D latents (Wang et al., 26 Feb 2026). The assignment cost is further regularized by an edge-based term, so the OT layer is both globally constrained and boundary-aware. The paper reports best geometry fidelity and second-best disentanglement among compared open-world scene-generation methods, and shows that removing OT plan-gated cross-attention worsens both geometry and disentanglement (Wang et al., 26 Feb 2026).

4. Structured spaces: heterogeneous modalities, graphs, and trees

When source and target live in different Euclidean spaces, the central difficulty is choosing an inter-space ground cost. Cost-regularized OT across spaces addresses this by learning a transform $F$9 jointly with the coupling: $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$0 and optimizing

$$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$1

The paper emphasizes sparsity, group sparsity / feature selection, and low rank as structure-inducing regularizers on $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$2, and gives a proximal alternating scheme in which the $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$3-step is solved by Sinkhorn and the $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$4-step has the proximal form

$$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$5

when $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$6 (Sebbouh et al., 2023). A central conceptual point is that many quadratic OT problems, including inner-product and squared-distance GW-like objectives, can be cast in this cost-regularized linear form.

For structured objects such as graphs, Fused Gromov-Wasserstein treats a graph as a weighted measure over node attributes and structural positions,

$$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$7

and combines feature costs with structural consistency through

$$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$8

The parameter $$\min_{\gamma \in M} f(\gamma)\equiv \min_{\gamma \in M}\max_{\kappa \in B_F}\langle \gamma,\kappa\rangle,$$9 interpolates between pure Wasserstein on features and pure Gromov-Wasserstein on structure (Vayer et al., 2018). For $B_F$0, FGW is a metric on structured data modulo measure-preserving isometries that preserve both features and structure; for $B_F$1, it is a semi-metric with relaxed triangle inequality factor $B_F$2. The paper also develops FGW barycenters and graph $B_F$3-means, thereby extending Fréchet means to graph-valued data (Vayer et al., 2018).

Tree-structured multi-marginal OT is another instance in which structure is embedded directly in the cost tensor. If the marginals correspond to nodes of a tree $B_F$4 and

$B_F$5

then the Gibbs kernel factorizes along edges: $B_F$6 This makes it possible to compute marginals by recursive message passing rather than explicit $B_F$7-tensor manipulation (Haasler et al., 2020). The paper proves that the entropy-regularized tree-structured MMOT problem is equivalent to a generalized Schrödinger bridge on the same tree, and argues that multi-marginal regularization introduces less diffusion than pairwise regularization because the objective includes extra negative entropy terms at internal nodes (Haasler et al., 2020).

5. Vector-valued, stochastic, and semimartingale generalizations

Some papers use “SOT” for formulations whose structure lies in admissible dynamics rather than in a modified static cost. Simultaneous Optimal Transport studies vector-valued measures

$B_F$8

and seeks a single map $B_F$9 or kernel $F$0 that satisfies all componentwise constraints simultaneously (Wang et al., 2022). The theory is explicitly different from classical OT: SOT is generally not symmetric, and the paper stresses that the dummy-point trick used to convert unbalanced scalar OT into balanced OT generally fails in this setting. Existence of simultaneous transport kernels is characterized by a multivariate increasing convex order condition on the laws of density vectors, and under joint atomlessness Monge and Kantorovich admissibility coincide (Wang et al., 2022). The same paper develops a Kantorovich-type dual with vector-valued potentials and introduces the MOT-SOT parity, a connection to a natural relaxation of martingale OT that yields explicit solutions in many special cases.

Stochastic optimal transport with fixed endpoint marginals treats transport as a controlled diffusion

$F$1

with value function

$F$2

For $F$3, $F$4, the paper shows that short-time behavior is governed by the Monge–Kantorovich cost $F$5; in particular,

$F$6

and boundedness of $F$7 is equivalent to $F$8 (Mikami, 2023). For costs with less than linear growth, the value is zero or infinity, and for $F$9 the framework recovers the Schrödinger problem.

Semimartingale Optimal Transport with jumps extends this pathwise viewpoint to laws on the Skorokhod space $F$0 with absolutely continuous differential characteristics $F$1 constrained to lie in a closed convex set $F$2. The transport cost is

$F$3

and the primal problem is

$F$4

Under measurability, convexity, lower semicontinuity, and compactness assumptions, the paper proves existence of optimal laws and a Kantorovich-type duality, and then shows that the same value admits four equivalent formulations: a dual problem, a viscosity-solution HJB problem, a martingale-solution formulation via Markovian projection, and a PDE formulation via a non-local Fokker–Planck–Kolmogorov equation (Hou et al., 25 Jun 2026). The framework is presented as simultaneously generalising classical OT, martingale OT, Schrödinger bridge problems, and barycentric weak OT.

The 2026 statistical-mechanics paper uses the acronym SOT for Sub-Optimal Transport rather than Structured Optimal Transport. It studies a finite-$F$5 ensemble of weighted bipartite graphs with exact strength constraints, shows that the dense-to-sparse change is a smooth crossover rather than a phase transition, and derives a large-$F$6 weight distribution $F$7 for uniform costs (Piombo et al., 4 Feb 2026). This terminological overlap is important because it can obscure substantive differences between structured and sub-optimal transport models.

6. Computation, statistical theory, and evolving boundaries of the field

A large part of the current literature is concerned with making structured OT computationally practical. In entropy-regularized semi-discrete OT, the source is continuous and the target discrete,

$F$8

and after eliminating the source potential the dual reduces to a convex minimization over target potentials,

$F$9

When the source density is discretized on a finite-element mesh, objective and gradient evaluation become dense source-to-target interaction problems of order $B_F$0 (Khamlich et al., 31 Jul 2025). The paper accelerates them by distance-based truncation of the Gibbs kernel, R-tree spatial queries, multilevel hierarchies on the source mesh and target measure, and $B_F$1-scheduling. Its implementation reports speedup of about $B_F$2 on 15 nodes, corresponding to roughly $B_F$3 parallel efficiency (Khamlich et al., 31 Jul 2025).

At the theoretical level, OT-based theory for latent structured models treats mixing measures and hierarchical random measures as the latent objects of interest and uses Wasserstein distances to connect latent geometry with observable data distributions. A forward inequality takes the form

$B_F$4

while the key structural result is an inverse bound of the form

$B_F$5

or, for finite mixtures,

$B_F$6

The paper argues that such inverse bounds encode identifiability and transfer estimation rates from the observed-data space to the latent space (Nguyen et al., 16 Jan 2026).

The boundaries of the term continue to expand. The 2025 review “An Introduction to Sliced Optimal Transport” explicitly states that in that review “structured” is used in a broad optimal-transport sense rather than as a single formalized “Structured OT” theory (Nguyen, 17 Aug 2025). There the structure comes from decomposing a high-dimensional problem into one-dimensional problems via Radon-type projections, weighted slicing, and geometry-aware transforms, yielding

$B_F$7

and computational cost

$B_F$8

for empirical measures (Nguyen, 17 Aug 2025). This suggests that the contemporary meaning of “structured OT” often includes decomposition strategies that preserve useful geometry while reducing computational and statistical complexity.

Across applications, the unifying pattern is not a single objective but a methodological stance: add structure where plain OT is too rigid, too noisy, too expensive, or too agnostic about relations among assignments. That stance underlies grouped and syntax-aware couplings in discrete OT, feature transforms that use test-time context, graph distances that fuse node features and topology, tree-structured Schrödinger bridges, cross-modal alignment with sparse or low-rank transforms, and path-space formulations constrained by semimartingale characteristics (Alvarez-Melis et al., 2017, Shalam et al., 2022, Vayer et al., 2018, Haasler et al., 2020, Sebbouh et al., 2023, Hou et al., 25 Jun 2026). A plausible implication is that the field will continue to be organized less by a single canonical model than by a shared question: which structural prior should be built into transport for a given data geometry, modality mismatch, or dynamical constraint.