Structure-Aware Optimal Transport

• SAOT is a family of optimal transport formulations that incorporate additional constraints beyond classical pairwise costs to enforce structure.
• It includes cost-, feasible-set-, and map-structured approaches that model coherence, causality, and topology through mechanisms like submodular costs and causal couplings.
• SAOT methods are applied in diverse fields such as graph learning, mesh processing, and cosmology, offering improved robustness and interpretability over traditional OT.

Structure-Aware Optimal Transport (SAOT) denotes a family of optimal transport formulations in which the transport plan or transport map is required to respect structure beyond the classical pairwise ground cost and marginal constraints. In the cited literature, that structure appears in several non-equivalent forms: nonlinear dependencies among assignment edges through submodular costs, hard order constraints on selected coupling entries, support masks that prohibit selected source–target matches, causal or bicausal admissibility on path space, topology- and orientation-preserving corrections for triangular meshes, and explicit preservation of relational organization in graph representations across tasks (Alvarez-Melis et al., 2017, Lim et al., 2021, Eckstein et al., 2022, Cang et al., 2022, Lv et al., 2 Jul 2025, Zhang et al., 1 Jul 2026). Taken together, these works indicate that SAOT is not a single canonical optimization problem, but a research program for injecting domain structure into OT at the level of the objective, feasible set, latent factorization, or computational oracle.

1. Classical OT and the SAOT viewpoint

Classical discrete Kantorovich OT solves

$$\min_{\Pi\in U(a,b)} \operatorname{tr}(D^\top \Pi),$$

where

$$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$

In this form, the transport cost is linear in the entries of the coupling, so assignments are modular: the cost of transporting mass from source index $i$ to target index $j$ is counted independently of what happens on other edges. The 2017 structured OT formulation identified this modularity as the central obstacle to modeling coherence, grouping, locality, or topological neighborhoods, and replaced the linear cost by a convex nonlinear functional derived from a submodular set function on assignment edges (Alvarez-Melis et al., 2017).

Other SAOT lines preserve the classical linear cost but alter admissibility. In adapted OT, couplings are restricted by temporal causality and bicausality on path space rather than by geometry alone (Eckstein et al., 2022). In supervised OT, selected source–target pairs are prohibited by assigning $+\infty$ cost and optimizing over plans in $U(\le a,\le b)$, with unmatched mass controlled by an exact $l^1$ penalty (Cang et al., 2022). In order-constrained OT, structure is encoded as linear inequalities that force selected coupling coefficients to dominate all others in a prescribed rank order (Lim et al., 2021). This suggests a useful editorial distinction: some SAOT models are cost-structured, some are feasible-set-structured, and some are map-structured.

The same broad perspective now appears in newer domain-specific work that explicitly uses the term SAOT. In continual graph learning, SAOT refers to preserving global inter-node correspondences across tasks rather than only stabilizing individual node embeddings, and optimal transport is used to maintain relational structure in representation space (Zhang et al., 1 Jul 2026).

2. Main formal mechanisms for injecting structure

The literature exhibits a small set of recurrent formal devices for making OT structure-aware.

Mechanism	Representative formulation	Representative work
Nonlinear dependence among assignments	$$\min_{\gamma\in M} f(\gamma)\equiv \min_{\gamma\in M}\max_{\kappa\in B_F}\langle \gamma,\kappa\rangle$$	Structured OT (Alvarez-Melis et al., 2017)
Hard ranking constraints on salient entries	$$\Pi_{i_k j_k}\ge \cdots \ge \Pi_{i_1 j_1}\ge \Pi_{pq}$$ for $(p,q)\in V$	Order-constrained OT (Lim et al., 2021)
Support masks and mass rejection	$$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$0, $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$1, $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$2-penalized unmatched mass	Supervised OT (Cang et al., 2022)
Temporal admissibility	$$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$3, $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$4 defined by causal or bicausal conditional-independence constraints	Adapted OT (Eckstein et al., 2022)
Relaxed mesh OT plus quasiconformal repair	relaxed semi-discrete OT followed by Beltrami-based local correction	QC-OT (Lv et al., 2 Jul 2025)

These mechanisms are mathematically distinct. In structured OT, the submodular set function $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$5 is lifted to its Lovász extension $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$6, giving a convex but non-separable transport objective. Classical OT is recovered when $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$7 is modular, so standard OT appears as a special case of the framework (Alvarez-Melis et al., 2017). In order-constrained OT, by contrast, the objective remains linear, but the feasible set is restricted by explicit linear inequalities over selected entries of the coupling; the result is convex whenever feasible (Lim et al., 2021).

Supervised OT shows a third pattern. Its lexicographic objective is to transport the largest possible amount of mass subject to support restrictions and then minimize cost on that maximal feasible mass. The paper proves equivalence, for sufficiently large $$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$8, to

$$U(a,b)=\left\{\Pi \in \mathbb{R}^{m\times n}_{+}:\ \Pi \mathbf{1}_n=a,\ \Pi^\top \mathbf{1}_m=b\right\}.$$9

which makes the structural supervision explicit as a support mask and an exact penalty on unmatched mass (Cang et al., 2022). Adapted OT uses neither nonlinear edge-set costs nor support masks. Instead, it restricts the admissible coupling class itself by causal identities such as

$i$0

yielding constrained transport over filtrations on path space (Eckstein et al., 2022).

QC-OT introduces a fourth pattern that is specific to map-based geometry processing. It preserves the original mesh connectivity by avoiding Delaunay retriangulation, then diagnoses and repairs local folding or excessive twisting by using the Beltrami coefficient of the inverse deformation. In that setting, structure means mesh connectivity, local orientation, and non-flip behavior, not merely mass balance (Lv et al., 2 Jul 2025).

3. Relational, graph, and low-rank structure

A separate branch of SAOT is organized around relational structure and tractability. In multimarginal OT, the decisive question is not only whether a cost tensor is “structured,” but whether one can efficiently solve the corresponding dual feasibility oracle. Exact polynomial-time solvability is characterized by

$i$1

approximate solvability by $i$2, and entropic Sinkhorn tractability by the soft oracle $i$3. This yields a unified account of graphical structure, set-optimization structure, and low-rank plus sparse structure in multimarginal OT, and it also shows that Sinkhorn requires strictly more structure than ellipsoid- or MWU-based methods (Altschuler et al., 2020).

Latent Optimal Transport (LOT) imposes low-rank structure by forcing transport through a small number of learned source anchors and target anchors. Its induced coupling factorizes through three stages—source points to source anchors, source anchors to target anchors, and target anchors to target points—and satisfies

$i$4

The method therefore encodes structure as an adaptive latent bottleneck, which the paper interprets as joint clustering and alignment; empirically it improves robustness to outliers, sampling noise, and source–target mismatch (Lin et al., 2020).

The Self-Optimal-Transport feature transform pushes this relational view inward, from transport between two distributions to self-coupling within a single set. Given one feature set, it computes a doubly stochastic self-transport matrix

$i$5

with forbidden diagonal transport, and uses the rows of $i$6 as new contextual embeddings. The paper’s interpretation is that transformed distances encode both direct similarity and “third-party agreement” regarding similarity to the rest of the set (Shalam et al., 2022). This is a self-relational form of SAOT rather than a cross-domain alignment problem.

In graph learning, the recent SAOT framework for self-supervised continual graph learning makes the relational agenda explicit. It computes a graph-space fused Gromov-Wasserstein plan $i$7 between augmented graph views, a representation-space OT plan $i$8 between learned embeddings, aligns the two, and then preserves structural knowledge across tasks by plan-level distillation. In the reported experiments, it improves average accuracy by up to $i$9 on CoraFull-CL and over $j$0 on Products-CL in the Class-IL setting (Zhang et al., 1 Jul 2026).

4. Temporal, sequential, and linguistic structure

Temporal structure enters SAOT through admissibility, chronology, or recursive decomposition. Adapted OT is the most direct example. It defines causal and bicausal transport problems

$j$1

with $j$2, and studies both LP approximation and entropic solvers. A principal algorithmic contribution is an adapted Sinkhorn method that alternates KL projections onto sets such as $j$3 and $j$4 or $j$5, with explicit backward-recursive projection formulas and linear convergence in objective value for the entropic problem (Eckstein et al., 2022). Here the structure is the information flow of a time series.

Spatio-Temporal Alignments (STA) combine OT-based spatial comparison with chronology-preserving temporal alignment. For each pair of time indices, the method computes an entropy-regularized unbalanced OT cost $j$6, debiases it into a Sinkhorn divergence $j$7, and then applies soft-DTW:

$j$8

The temporal structure is enforced by monotone DTW paths, while spatial structure is handled by unbalanced OT on each time slice. The paper proves that soft-DTW increases quadratically with time shifts under explicit conditions, which distinguishes STA from chronology-blind OT on flattened spatio-temporal data (Janati et al., 2019).

In natural language processing, Recursive Optimal Transport (ROT) makes sentence similarity sensitive to word order and syntax by replacing a flat bag-of-words OT problem with a hierarchy of OT problems over recursively defined substructures. The central recursion uses a KL-regularized OT problem with a prior inherited from the previous coarser level, so higher-level alignments constrain lower-level ones. The framework admits binary trees built from word order as well as dependency trees, and its similarity version, ROTS, aggregates transport-weighted similarities across levels (Wang et al., 2020). This is a tree-structured SAOT framework rather than a generic OT solver.

5. Geometric and topological structure

The most explicit topology-aware SAOT formulation in the cited material is QC-OT for triangular meshes. The paper starts from the Monge-map viewpoint for quadratic cost, then uses semi-discrete geometric OT as the computational baseline. Its criticism of standard semi-discrete OT is specific: Delaunay triangulation updates preserve convexity and existence of the Brenier solution, but they can change mesh connectivity; if one restores the original connectivity afterward, skinny or degenerate triangles can result. QC-OT therefore freezes the original triangulation, relaxes the admissible set from strict positivity of power-cell mass to nonnegativity,

$j$9

and accepts weaker convexity guarantees in exchange for preserving combinatorial topology (Lv et al., 2 Jul 2025).

After the relaxed semi-discrete OT stage, QC-OT computes the inverse deformation $+\infty$0 and diagnoses distortion by the Beltrami coefficient $+\infty$1, using the criterion that $+\infty$2 corresponds to a diffeomorphic quasiconformal map. Regions with $+\infty$3 or vertexwise $+\infty$4 are repaired locally on $+\infty$5-ring patches, with $+\infty$6 in the experiments. The final pipeline preserves original mesh connectivity by construction and seeks local orientation preservation and non-folding through quasiconformal correction. The paper also defines a temporal extension, tt-OT, as repeated relaxed Brenier optimization plus QC correction over iterations, producing topology-preserving intermediate meshes. The intended applications are mesh parameterization, image editing, image-driven mesh generation, and medical image magnification, while the stated limitations include specialization to triangular meshes, simply connected domains, 2D parameter domains for 3D surfaces, and weaker convergence guarantees after relaxing convexity (Lv et al., 2 Jul 2025).

A broader physical interpretation of geometry-aware transport appears in cosmology. The paper on Wasserstein distance in cosmological structure formation treats the mapping from an initial continuous density field to an observed galaxy point process as a hierarchical transport problem and derives an approximate decomposition

$+\infty$7

The gravitational term is

$+\infty$8

the galaxy contribution appears as a transport-weighted integral involving $+\infty$9, and the sampling term is Poisson shot noise. This formulation treats Wasserstein geometry as a statistic of spatial redistribution rather than only fluctuation amplitude, which is a structurally aware interpretation of OT on continuous-to-discrete generative hierarchies (Takeuchi, 16 Mar 2026).

6. Algorithms, applications, and limitations

Algorithm design in SAOT is inseparable from the chosen structural prior. Submodular structured OT uses mirror descent or mirror-prox on the transport polytope together with greedy subgradient computation on the base polytope (Alvarez-Melis et al., 2017). Order-constrained OT splits the problem into projections onto the marginal set and the order-constrained set and solves it by ADMM, with the nontrivial projection implemented by the specialized ePAVA procedure (Lim et al., 2021). Curriculum and Structure-aware OT for noisy-label learning adds nonconvex local-coherence terms to a sample-to-class OT allocator and uses generalized conditional gradient outside a Sinkhorn-like scaling solver for the entropic curriculum subproblem (Chang et al., 2023). Support-supervised OT uses generalized Sinkhorn or Bregman-Dykstra iterations with clipped marginal updates induced by the $U(\le a,\le b)$0-penalized inequality constraints (Cang et al., 2022).

The application range is correspondingly broad. The cited papers cover domain adaptation and syntax-aware sentence similarity (Alvarez-Melis et al., 2017), explainable natural-language alignment and image color transfer (Lim et al., 2021), surface mesh parameterization and medical image magnification (Lv et al., 2 Jul 2025), noisy-label denoising and relabeling (Chang et al., 2023), few-shot classification, clustering, and person re-identification (Shalam et al., 2022), continual graph learning (Zhang et al., 1 Jul 2026), and cosmological large-scale structure formation (Takeuchi, 16 Mar 2026). A plausible implication is that SAOT is most useful where the object of interest is not just a pair of marginals, but a pair of structured representations whose admissible correspondences are constrained by causality, topology, hierarchy, local coherence, or a latent relational summary.

Several misconceptions recur. SAOT is not synonymous with entropic regularization or with “regularized OT” in general: the cited works show structure entering through nonlinear edge-set costs, hard order or support constraints, causal couplings, anchor factorizations, or post-transport geometric correction. Nor is there a universally dominant solver. In multimarginal OT, Sinkhorn requires efficient $U(\le a,\le b)$1 evaluation and is strictly less universal than methods based on $U(\le a,\le b)$2 or $U(\le a,\le b)$3 (Altschuler et al., 2020). Feasibility is also often delicate: order constraints require nonempty structured coupling sets (Lim et al., 2021), support masks may force mass rejection (Cang et al., 2022), and QC-OT explicitly weakens convexity and can exhibit slower convergence, instability, or oscillation after power-cell relaxation (Lv et al., 2 Jul 2025).

The limitations are therefore structural as well as computational. Many SAOT models are specialized: adapted OT is tailored to finite-horizon path spaces (Eckstein et al., 2022), QC-OT to simply connected triangular meshes and parameter domains (Lv et al., 2 Jul 2025), LOT to low-rank anchor-compressible geometry (Lin et al., 2020), and support-supervised OT to pairwise admissibility masks rather than richer relational priors (Cang et al., 2022). The field’s unifying theme is nonetheless clear. SAOT extends OT by encoding which correspondences, deformations, or relational organizations are permissible, stable, or interpretable in a given domain, and then redesigning both theory and computation around that structural choice.