Inverse Optimal Transport

• Inverse optimal transport is a framework that infers hidden cost matrices or structures from observed couplings by reversing the standard OT formulation.
• It employs regularization methods such as entropy and Bregman approaches to address non-identifiability and enhance computational efficiency.
• The approach is applied in domains like matching markets, migration, and signal processing, using both deterministic and Bayesian strategies.

Searching arXiv for recent and foundational papers on inverse optimal transport to ground the article. Inverse optimal transport denotes a family of inverse problems in which the forward optimal transport map is reversed: instead of prescribing a cost and solving for an optimal coupling, one observes transport outputs—such as a coupling, a noisy matching matrix, optimal values, or transport maps—and seeks the latent cost or related structure that would make those observations optimal. In its canonical discrete form, the problem is to infer a ground cost matrix $C$ from an observed coupling $\hat\pi$; in broader formulations it includes recovery of latent marginals, transition laws, or geometric structure from OT observables (Li et al., 2018, Stuart et al., 2019, Ma et al., 2020, González-Sanz et al., 2023, Zhai et al., 19 Nov 2025). The subject sits at the intersection of OT, inverse problems, statistical inference, and matching theory, and its central technical themes are non-identifiability, regularization, and computational tractability.

1. Emergence and scope

Inverse OT developed first around the question of recovering an unknown matching or transport cost from observed matchings. In two-sided matching markets, the motivating observation is that empirical matching matrices may be viewed as the outcome of an underlying optimization principle, so one may ask which cost function would make the observed matching optimal (Li et al., 2018). A closely related 2019 formulation treated noisy transport plans as observations of a discrete Kantorovich problem and embedded the inference of costs, sources, and targets into a Bayesian inverse-problem framework (Stuart et al., 2019). By 2020, inverse OT had also been reformulated as a single-level convex variational problem for entropy-regularized OT, emphasizing that the classical bilevel structure need not be solved by repeatedly running a forward solver from scratch (Ma et al., 2020).

The scope of the field subsequently widened along several axes. One line emphasized probabilistic inference and ambiguity, arguing that the inverse target is often not a unique cost matrix but an equivalence class or posterior over costs compatible with the observed coupling (Chiu et al., 2021). Another line shifted the observed data from plans to optimal values and marginals, yielding a nonlinear inverse OT problem in which identifiability is mediated by optimal dual potentials rather than by the transport plan alone (González-Sanz et al., 2023). On finite spaces, the subject became a question of polyhedral identifiability governed by the combinatorics of transport polytopes (González-Sanz et al., 2024). On smooth continuous domains, later work argued that regularity of the marginals fundamentally changes the inverse problem by introducing genuine curvature with respect to the cost, leading to local identifiability and stability modulo intrinsic invariances (Peyré et al., 24 Apr 2026).

The term is also used in adjacent but non-identical ways. Some works formulate inverse problems whose unknowns are not transport costs but dynamical laws, metrics, or source measures, while still using OT structure centrally; examples include Markov chain estimation from aggregate distributions, recovery of a Riemannian metric from OT maps, and OT-based stability for inverse point-source problems (Mascherpa et al., 20 Nov 2025, Zhai et al., 19 Nov 2025, Qiu et al., 26 Dec 2025). This suggests that “inverse optimal transport” now names both a core cost-recovery problem and a broader class of inverse problems in which OT observables are the data.

2. Canonical formulations

The forward problem is the discrete Kantorovich program. Given marginals $\mu$ and $\nu$, the feasible couplings form

$$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$

and the transport value is

$$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$

For entropy-regularized OT, one instead minimizes

$\langle c,\pi\rangle-\varepsilon H(\pi),$

or, in equivalent notation used across the literature, $\langle c,\pi\rangle+\epsilon H(\pi)$ depending on sign convention for the entropy term (Li et al., 2018, Stuart et al., 2019, Ma et al., 2020). The optimizer then has the Sinkhorn form

$$\pi^\star=\operatorname{diag}(u)\,K\,\operatorname{diag}(v),\qquad K_{ij}=e^{-c_{ij}/\varepsilon},$$

which is one reason entropic OT dominates the computational side of inverse OT.

In the basic inverse problem, one observes $\hat\pi\in U(\mu,\nu)$ and seeks a cost $\hat\pi$0 such that the forward plan $\hat\pi$1 induced by $\hat\pi$2 resembles $\hat\pi$3. A standard entropic formulation minimizes a divergence such as

$\hat\pi$4

possibly with a regularizer on $\hat\pi$5 (Li et al., 2018, Ma et al., 2020). The 2020 convex reformulation writes the inverse problem directly as an unconstrained convex problem over dual variables $\hat\pi$6 and the cost: $\hat\pi$7 where $\hat\pi$8 contains linear terms in $\hat\pi$9 and a convex exponential term inherited from the entropic dual (Ma et al., 2020). This eliminated the need for a full forward OT solve inside every outer iteration.

Several extensions alter the observed object or the structural hypothesis on the cost. In matching, RIOT replaces hard marginal constraints by latent marginals $\mu$0 penalized through regularized Wasserstein distances to empirical marginals, thereby learning a nonlinear kernelized interaction cost $\mu$1 from noisy, incomplete, or biased matching data (Li et al., 2018). In Bayesian inverse OT, the observation model is

$\mu$2

with $\mu$3 the forward OT map and $\mu$4 Gaussian noise, so the posterior over latent marginals and costs is sampled rather than optimized deterministically (Stuart et al., 2019). In Bregman-regularized inverse OT, the forward problem becomes

$\mu$5

with a Legendre-type Bregman generator $\mu$6, allowing entropy, Burg entropy, Fermi–Dirac entropy, and $\mu$7-potentials within a common inverse theory (Bao et al., 4 Oct 2025).

A further generalization appears when the observed data are not couplings but marginal pairs and optimal values. For translation-invariant costs $\mu$8, the inverse problem becomes nonlinear: the aim is to identify $\mu$9 from a sufficiently rich set of marginals together with the corresponding optimal values $\nu$0 (González-Sanz et al., 2023). In yet another direction, Markov chain estimation from aggregate distributions is cast as a KL-regularized inverse OT problem in which the unknowns are both transport plans $\nu$1 and a common row-stochastic transition matrix $\nu$2, with the latter recovered from the aggregate transport $\nu$3 (Mascherpa et al., 20 Nov 2025).

3. Identifiability, ambiguity, and equivalence classes

Non-identifiability is the defining structural issue of inverse OT. In the entropic discrete setting, only the combination $\nu$4 matters for the induced coupling, so many triples $\nu$5 generate the same plan (Ma et al., 2020). A standard invariance is the additive transformation

$\nu$6

which leaves the set of optimal couplings unchanged (Ma et al., 2020, González-Sanz et al., 2024, Peyré et al., 24 Apr 2026). On finite spaces this becomes shift equivalence $\nu$7, and any recovery from plans alone can at best identify the cost in the quotient space by that shift subspace (González-Sanz et al., 2024). Scaling invariance is also intrinsic in several formulations; in smooth continuous bilinear models the admissible set is often identifiable only up to the positive ray $\nu$8 (Peyré et al., 24 Apr 2026).

The probabilistic discrete theory sharpened this point by introducing cross-ratio equivalence. For $\nu$9, two costs are equivalent if the kernels have the same $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$0 cross-ratios, or equivalently if all discrete second differences of the costs agree (Chiu et al., 2021). The resulting inverse target is therefore a manifold of cross-ratio equivalent costs rather than a unique matrix. The same paper argued that priors are not a technical afterthought but the mechanism that selects representatives from this manifold (Chiu et al., 2021). The 2023 sparsistency analysis made the same ambiguity visible from a different angle: even though entropic inverse OT is convex, support recovery for a sparse parameterization still requires an $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$1-penalized estimator and a generalized irrepresentability condition analogous to the Lasso (Andrade et al., 2023).

Finite-space identifiability admits a precise polyhedral characterization. The observed information may consist of total costs, plans, potentials, or combinations thereof, and each regime yields necessary-and-sufficient uniqueness criteria in terms of faces and extreme points of the transport polytope (González-Sanz et al., 2024). The strongest exact result is the full-information case: if total costs, plans, and optimal potentials are observed, then the cost is identifiable exactly when the union of the supports of the observed optimal plans covers the entire matrix (González-Sanz et al., 2024). By contrast, if only plans are observed, uniqueness is inherently modulo shift equivalence.

Continuous theories partly reverse the pessimism of the discrete setting. For nonlinear inverse OT from marginals and optimal values, the cost is identified on the union of the ranges of the gradients of the optimal potentials; when an open set of marginals is observed, the optimal values can identify the dual potentials, and hence the cost, up to an additive constant under suitable convex or concave structural assumptions (González-Sanz et al., 2023). In the smooth continuous bilinear setting, the second variation of the OT value with respect to the cost is non-degenerate modulo natural transport invariances, yielding a strict curvature property absent in discrete OT and implying local identifiability and stability under a spanning condition on the Hessians of the Brenier potential (Peyré et al., 24 Apr 2026). This suggests a sharp dichotomy: discrete inverse OT is generically flat and polyhedral, whereas sufficiently regular continuous inverse OT can be curved and locally invertible.

4. Algorithms and computational strategies

Most computational schemes exploit the entropic or Bregman regularity of the forward problem. RIOT solves a non-convex objective by alternating optimization over the interaction matrix and latent marginals on one side, and dual variables associated with the relaxed marginal penalties on the other; Sinkhorn-Knopp scaling is used inside both steps, and the gradient with respect to the interaction matrix is obtained by the envelope theorem (Li et al., 2018). The method was designed specifically for noisy, incomplete, and biased matching matrices, where exact empirical marginals are treated as unreliable (Li et al., 2018).

Bayesian inverse OT replaces deterministic descent by posterior sampling. Its Random Walk Metropolis within Gibbs algorithm perturbs $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$2, $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$3, and $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$4 or a structured parameter $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$5 blockwise, accepts proposals according to the change in data misfit, and requires only repeated forward OT solves plus random-number generation (Stuart et al., 2019). Because the forward solver can be either exact linear programming OT or entropic Sinkhorn OT, the method separates the statistical formulation from the particular OT engine (Stuart et al., 2019). Its principal advantage is uncertainty quantification rather than speed.

Single-level convex reformulations enable faster deterministic solvers. In discrete entropy-regularized inverse OT, the 2020 matrix-scaling method alternates updates of Sinkhorn-type scaling vectors and a proximal update for the cost, thereby avoiding a full forward OT solve in each outer iteration (Ma et al., 2020). The same paper also introduced a mesh-free continuous method in which the cost and dual potentials are parameterized by neural networks and trained by stochastic gradient descent on empirical samples from $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$6, $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$7, and $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$8 (Ma et al., 2020). The Bregman-regularized extension later proposed an inexact block coordinate descent method for the single-level objective, with linear convergence under a strongly convex penalty term; in the quadratic-penalty case the subproblems have diagonal Hessian structure, enabling efficient element-wise Newton updates (Bao et al., 4 Oct 2025).

Convexity reappears in more specialized models. For Markov chain estimation from aggregate snapshots, the inverse OT problem over transport plans and a common transition matrix is jointly convex; after eliminating the transition matrix, the remaining optimization over the couplings and their aggregate transport admits an entropic proximal iterative scheme whose subproblems are standard Sinkhorn-solvable OT problems (Mascherpa et al., 20 Nov 2025). In supervised inverse partial OT for music-guided trailer generation, learning takes the form of a bilevel program: the lower level is an entropic partial OT problem solved by a differentiable Sinkhorn network, and the upper level fits encoder parameters so that the induced alignment plan matches annotated movie–trailer correspondences (Wang et al., 2024). These variants illustrate a recurring pattern: convexity or differentiability is typically obtained by entropic or KL-type regularization, while exact unregularized inverse OT remains substantially harder.

5. Domains of application

The application literature uses inverse OT when the observed object is a coupling or origin–destination flow and the latent scientific object is a cost, affinity, or transition law. In matching markets, the framework is explicitly prescriptive: it aims not only to predict future matching but also to explain observed matching and quantify the effect of changing matching factors (Li et al., 2018). In migration, border-graph inverse OT interprets the observed flow matrix as a noisy transport plan and estimates edge weights that act as “migration transition costs” together with posterior uncertainty (Stuart et al., 2019).

Domain	Inverse OT object	Representative paper
Two-sided matching and marriage	Interaction cost or affinity matrix	(Li et al., 2018, Ma et al., 2020, Bao et al., 4 Oct 2025)
International migration	Graph edge weights and latent marginals	(Stuart et al., 2019)
Aggregate Markov dynamics	Common transition matrix from couplings	(Mascherpa et al., 20 Nov 2025)
Movie trailer generation	Cross-modal grounding distance and partial OT plan	(Wang et al., 2024)
Urban access and school choice	Latent access cost from origin–destination flows	(Martinez, 12 Jun 2026)

The empirical claims in these domains are varied but concrete. RIOT was evaluated on synthetic data, Florida 2012 election data, New York Taxi data, and a marriage data set; on the Dutch Household Survey couples, it outperformed Random, SVD, itemKNN, PMF, and Factorization Machines, with RMSE $$U(\mu,\nu)=\{\pi\ge 0:\pi \mathbf 1=\mu,\ \pi^\top \mathbf 1=\nu\},$$9 and MAE $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$0 (Li et al., 2018). The 2020 convex inverse OT paper also reported marriage-market results on the Dutch Household Survey, where the learned affinity matrix was interpretable and the method achieved the best accuracy with much faster runtime than RIOT and standard recommender baselines (Ma et al., 2020). In the migration network example with nine countries and 30 border edges, exact OT and Sinkhorn-based inverse posteriors were reported to be similar, with acceptance rates around $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$1 for exact OT and $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$2 for Sinkhorn in the reported run (Stuart et al., 2019).

Urban-access estimation provides a more policy-facing example. Treating school-to-school enrollment flows in the Philippines as an entropic OT plan, a 2026 study estimated a distance-banded access cost with a subsidy term and interpreted the ratio

$$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$3

as subsidy-equivalent distance in kilometers per 1,000 pesos (Martinez, 12 Jun 2026). On $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$4 learner trips across $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$5 observed flows, the estimated values were $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$6 km, $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$7 km, and $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$8 km (Martinez, 12 Jun 2026). In media generation, inverse partial OT was used to learn the grounding distance between movie shots and trailer music shots from aligned movie–trailer data; the resulting CMTD dataset contains 208 movies and 406 trailers, and the method reported Selection F1@1/F1@3/F1@5 of $$T_c(\mu,\nu)=\min_{\pi\in U(\mu,\nu)} \langle c,\pi\rangle.$$9, $\langle c,\pi\rangle-\varepsilon H(\pi),$0, and $\langle c,\pi\rangle-\varepsilon H(\pi),$1, with Alignment F1@1 $\langle c,\pi\rangle-\varepsilon H(\pi),$2 and KL divergence $\langle c,\pi\rangle-\varepsilon H(\pi),$3 (Wang et al., 2024). These applications indicate that inverse OT has become a method for extracting latent behavioral or structural costs from aggregate correspondences rather than only a theoretical inverse problem.

6. Boundaries of the concept, open issues, and related directions

Several limitations recur across the literature. The inverse problem may be underdetermined, especially for general cost matrices or high-dimensional parameterizations (Stuart et al., 2019, González-Sanz et al., 2024). Entropic regularization improves smoothness and numerical stability but introduces bias or blurring, and the regularization parameter is itself non-identifiable from the inverse problem in the entropy-regularized setting because only the ratio $\langle c,\pi\rangle-\varepsilon H(\pi),$4 is recoverable (Ma et al., 2020). Sparse recovery depends on a non-degenerate certificate and can deteriorate as $\langle c,\pi\rangle-\varepsilon H(\pi),$5 varies, while finite-sample theory carries explicit exponential dependence on $\langle c,\pi\rangle-\varepsilon H(\pi),$6 in the reported rates (Andrade et al., 2023). Even in Bregman-regularized settings, exact uniqueness requires structural assumptions such as symmetry with zero diagonal, metric structure, or Euclidean distance constraints; otherwise one recovers only an equivalence class (Bao et al., 4 Oct 2025).

A second frontier concerns the relation between inverse OT proper and the wider use of OT in inverse problems. Dynamic inverse problems in spaces of measures use OT as a regularizer rather than as an unknown to be inferred; the unknown is a time-dependent measure curve constrained by a continuity equation, and the OT term is a dynamical prior (Bredies et al., 2019). Generalized Sinkhorn iterations have likewise been used to compute proximal operators for OT-regularized inverse problems in limited-angle CT, where the objective is reconstruction with an OT prior rather than recovery of a transport cost (Karlsson et al., 2016). OT-CycleGAN for unsupervised inverse problems derives a cycle-consistent adversarial architecture from Kantorovich duality and a penalized least-squares transport cost, but its goal is learning an inverse map $\langle c,\pi\rangle-\varepsilon H(\pi),$7, not identifying a latent OT cost from a coupling (Sim et al., 2019). These distinctions mark the boundary between inverse OT as a cost-recovery problem and OT-informed inverse problems more broadly.

The field has also moved toward geometric inverse questions outside classical cost learning. One 2025 result proves that on a compact closed Riemannian manifold, the family of optimal maps for the quadratic cost $\langle c,\pi\rangle-\varepsilon H(\pi),$8 determines the underlying metric up to a multiplicative constant (Zhai et al., 19 Nov 2025). Another establishes OT-based stability estimates for inverse elliptic and parabolic point-source problems by realizing Kantorovich dual potentials as boundary functionals of adjoint PDE solutions (Qiu et al., 26 Dec 2025). A plausible implication is that inverse OT is becoming a general language for inverse problems in which transport geometry, duality, and observability interact. The persistent challenge is to distinguish intrinsic ambiguity from recoverable structure: where discrete models expose flat cones and equivalence classes, smooth continuous models show that regularity, excitation, and structural constraints can restore curvature, stability, and local uniqueness (Peyré et al., 24 Apr 2026).