Inverse Optimal Transport (IOT) Overview

• IOT is a framework that recovers latent ground cost functions from observed optimal transport outputs like couplings, transport maps, and potentials.
• It leverages entropic and Bregman regularization to convert ill-posed inverse problems into convex, scalable formulations.
• The approach distinguishes discrete non-identifiability up to additive invariances from continuous settings that achieve local uniqueness through strict curvature.

Inverse Optimal Transport (IOT) studies the inverse map from optimal-transport observables—optimal couplings, transport maps, Kantorovich potentials, or optimal values—back to the latent ground cost or related structure that renders those observables optimal. In discrete settings, IOT is closely related to inverse linear programming and is intrinsically ill-posed; under entropic or more general Bregman regularization, it admits convex formulations and scalable solvers; in smooth continuous settings, regularity of the marginals can create strict curvature of the optimal-transport functional with respect to the cost, yielding local identifiability modulo the standard transport invariances (González-Sanz et al., 2023, Chiu et al., 2021, Peyré et al., 24 Apr 2026).

1. Formal scope and canonical formulations

In classical optimal transport, one starts from marginals $\mu,\nu$ and a known ground cost $c$, and solves the Kantorovich problem

$$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$

or its entropically regularized analogue. IOT reverses this direction: the cost is unknown, while some OT-related object is observed and used to recover $c$ or its parametrization (González-Sanz et al., 2023).

The observed object depends on the regime. A standard discrete formulation assumes known marginals and an observed coupling $\hat\pi$, and asks for all costs $C$ such that $\hat\pi$ is the optimizer of entropy-regularized OT. In that setting, the forward solution has Schrödinger form

$\pi^*_{ij}(C)=a_i b_j e^{-C_{ij}/\varepsilon},$

so inverse recovery is governed by how $C$ enters this exponential family (Chiu et al., 2021). A different line of work studies nonlinear IOT, where one does not observe optimal plans at all, but only a family of marginals $(\mu_\theta,\nu_\theta)$ together with the scalar optimal values

$c$0

In that regime the inverse map $c$1 is genuinely nonlinear, and identifiability depends on the structure of the admissible cost class and the richness of the observed marginals (González-Sanz et al., 2023).

The inverse variable is not always a free cost matrix. Several works constrain the unknown to a parametric family such as $c$2, translation-invariant convex costs $c$3, concave distance costs $c$4, shortest-path costs on graphs, or squared Riemannian distance generated by an unknown metric tensor (González-Sanz et al., 2023, Stuart et al., 2019, Zhai et al., 19 Nov 2025). This makes IOT a family of inverse problems rather than a single model class.

2. Non-identifiability, invariances, and conditions for identifiability

A central feature of IOT is non-identifiability. In discrete entropic OT, if $c$5 is strictly positive, every compatible cost has the form

$c$6

so costs are identifiable only up to additive row and column terms. Equivalently, two costs generate the same optimal plan if and only if they are cross-ratio equivalent, meaning they have identical quantities

$c$7

for every $c$8 submatrix. The resulting solution set is an affine manifold of dimension $c$9 (Chiu et al., 2021).

This discrete ill-posedness has a broader inverse-linear-programming interpretation. In a finite OT problem, the feasible set of inverse costs is typically a polyhedral cone determined by the normal cone of the observed optimizer. For bilinear cost families, unless the effective rank induced by the support of the plan is maximal, the feasible parameter set has dimension at least $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$0; flat directions are therefore structural rather than numerical artifacts (Peyré et al., 24 Apr 2026).

Smooth continuous OT changes this picture. When source and target marginals have smooth positive densities on uniformly convex domains, and the reference cost satisfies a strong twist condition, the map $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$1 is $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$2 in Hölder spaces. Its second variation is non-degenerate modulo the natural invariances $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$3, and vanishes only along those invariant directions. This strict curvature yields local identifiability and stability for inverse OT that are absent in discrete transport (Peyré et al., 24 Apr 2026).

For the structured family $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$4, the continuous theory becomes concrete. If the Brenier map for the true cost has sufficiently rich curvature, expressed through a spanning condition on the Hessians of the associated convex potential, then the feasible set of matrices collapses to the positive ray $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$5. In that case a single optimal coupling identifies the Mahalanobis parameter up to the unavoidable scaling invariance. By contrast, when the OT map is affine—as for Gaussian or more general elliptical marginals with the same base—this spanning condition fails and identifiability remains degenerate (Peyré et al., 24 Apr 2026).

Nonlinear IOT from scalar optimal values has a distinct identifiability mechanism. For convex translation-invariant costs $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$6, the cost is determined by the union of the ranges of the gradients of the optimal potentials; when an open set of marginals is observed, sensitivity analysis of the OT value functional identifies the potentials, and hence $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$7, up to an additive constant. In one dimension, the problem reduces to an integral transform of $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$8, and identifiability becomes equivalent to statistical completeness of the underlying location-scale family (González-Sanz et al., 2023).

3. Regularization and geometric structure

Entropic regularization is the most common analytic device in IOT. It makes the forward OT plan strictly positive and differentiable with respect to the cost, which both improves computation and weakens the severe ambiguity induced by sparse unregularized plans. Inverse OT can then be written as a KL-fitting problem between an observed coupling $$\mathcal{T}_c(\mu,\nu) := \inf_{\pi \in \Pi(\mu,\nu)} \int c(x,y)\, d\pi(x,y),$$9 and the induced entropic OT plan $c$0, possibly with extra regularization on $c$1 (Ma et al., 2020).

A particularly important consequence is that the inverse problem can be written as a single-level convex program. For continuous entropy-regularized OT, one can optimize jointly over $c$2 using

$c$3

augmented by any convex regularizer $c$4. This eliminates the explicit bilevel structure of “outer inverse, inner OT” and avoids repeatedly solving a forward OT problem inside each inverse step (Ma et al., 2020).

Bregman regularization generalizes the entropic case. For discrete OT with regularizer $c$5, where $c$6 is a strictly convex Legendre-type generator applied entrywise, the forward map $c$7 remains single-valued, and the inverse problem admits a single-level convex reformulation in terms of the Fenchel conjugate $c$8: $c$9 Under the condition $\hat\pi$0, this framework yields existence, uniqueness up to equivalence classes, and stability results for several structured cost sets, including symmetric zero-diagonal costs and Euclidean distance matrices (Bao et al., 4 Oct 2025).

Regularization also shapes statistical structure. In sparse entropic IOT with linearly parameterized cost $\hat\pi$1, adding an $\hat\pi$2 penalty produces a Lasso-type inverse problem. The corresponding sparsistency theory shows that support recovery depends on a generalized irrepresentability condition involving the Hessian $\hat\pi$3 of the entropic OT log-likelihood. In Gaussian settings, the problem interpolates between a classical Lasso regime as $\hat\pi$4 and a graphical Lasso regime as $\hat\pi$5 (Andrade et al., 2023).

Unbalanced OT provides another geometric extension. In unpaired inverse problems, replacing exact marginal constraints by KL penalties on marginal mismatch yields an unbalanced OT map with cost

$\hat\pi$6

The likelihood term encodes data fidelity, while the quadratic term restores the twist condition and hence existence and uniqueness of the OT map when the forward operator is ill-posed (Lee et al., 20 May 2026). A plausible implication is that inverse OT and inverse-problem regularization can be unified at the level of cost design rather than only at the level of penalties.

4. Computational methods

Computationally, IOT splits into two broad families: methods that differentiate through a forward OT solver, and methods that optimize a direct inverse objective without repeatedly solving the full forward problem.

The first family is dominant in applications. In entropy-regularized discrete OT, the forward map is implemented by Sinkhorn scaling with Gibbs kernel $\hat\pi$7, and inverse learning updates cost parameters by backpropagating through the Sinkhorn iterations. This appears explicitly in legal case matching, where a bi-level KL objective fits the OT plan to sparse sentence alignments (Yu et al., 2022); in inverse partial OT for music-guided trailer generation, where Sinkhorn operates on a selected subset of source shots (Wang et al., 2024); and in urban access-cost estimation, where a differentiable Sinkhorn layer is used to fit school-to-school flows under both a distance-banded model and a neural cost model (Martinez, 12 Jun 2026).

The second family exploits convex reformulations. For entropy-regularized inverse OT, one can optimize the convex objective $\hat\pi$8 directly. In the discrete case this leads to a matrix-scaling algorithm that alternates between Sinkhorn-type updates for $\hat\pi$9 and a proximal update for $C$0, avoiding a nested forward solve at every outer iteration (Ma et al., 2020). For Bregman-regularized IOT, the single-level convex objective with a strongly convex penalty is amenable to exact or inexact block coordinate descent. When the penalty is quadratic, the Hessians of the block subproblems are diagonal, so the Newton steps reduce to elementwise updates. The resulting inexact BCD algorithm has a linear convergence rate (Bao et al., 4 Oct 2025).

Bayesian and probabilistic formulations constitute a third computational style. In discrete probabilistic IOT, the cost matrix is treated as a latent random variable and the posterior is sampled by Metropolis–Hastings, with each likelihood evaluation requiring an entropy-regularized OT solve. This framework explicitly represents posterior uncertainty over the cross-ratio equivalence class rather than collapsing it to a single point estimate (Chiu et al., 2021). A related Bayesian inverse-OT formulation for migration flows infers graph edge costs from noisy transport plans, again using forward OT solvers as black boxes inside MCMC (Stuart et al., 2019).

Convexity can also emerge in unexpected inverse-OT variants. In aggregate Markov-chain estimation, one jointly infers transport plans $C$1 and a common transition matrix $C$2 by minimizing

$C$3

which is equivalent to a convex problem in $C$4 once the aggregate prior flow $C$5 is introduced. The associated entropic proximal algorithm reduces each iteration to standard Schrödinger-bridge or entropic OT subproblems (Mascherpa et al., 20 Nov 2025).

5. Variants and applications

IOT has developed along several application-driven branches. In economics and mobility, graph-based inverse OT has been used to infer latent migration costs from observed international migration flows, where edge weights on a border graph determine pairwise transport costs through shortest paths (Stuart et al., 2019). In matching markets, feature-based inverse OT learns preference or affinity matrices that explain marriage patterns, labor matching, or similar two-sided assignments; robust variants introduce marginal relaxation through regularized Wasserstein terms to handle noisy empirical marginals (Li et al., 2018).

In machine learning, IOT increasingly appears as a differentiable alignment layer. Legal case matching formulates sentence-level rationale extraction as inverse OT over an entropy-regularized alignment matrix; the learned transport plan is then interpreted directly as an explanation map (Yu et al., 2022). Music-guided trailer generation uses inverse partial OT to learn a cross-modal ground cost between movie shots and music shots, with only a subset of movie shots participating in the transport (Wang et al., 2024). Urban school-choice analysis treats observed school-to-school enrollment flows as an entropic OT plan and learns latent access costs using both an interpretable distance-banded model and a neural cost model (Martinez, 12 Jun 2026).

A broader inverse-problem perspective has also emerged. On closed Riemannian manifolds, knowledge of all optimal maps for the squared geodesic-distance cost determines the unknown metric tensor up to a global multiplicative constant. After linearization around the identity transport, the problem reduces to an inverse coefficient problem for an elliptic PDE (Zhai et al., 19 Nov 2025). In elliptic and parabolic inverse source problems, OT enters as the metric on the space of sources rather than as the forward model; the Kantorovich dual potentials are represented by adjoint boundary functionals, yielding stability estimates of the form

$C$6

and its parabolic analogue (Qiu et al., 26 Dec 2025). These works do not learn the cost, but they extend inverse-OT methodology by importing OT geometry into inverse PDE analysis.

Inverse reinforcement learning provides another adjacent variant. Reward ambiguity can be treated geometrically by viewing the set of reward functions consistent with an expert policy as a subset of a Wasserstein space and defining a centroid

$C$7

This is not a canonical OT inverse problem on couplings, but it is inverse-OT in spirit because OT geometry is used to resolve multiplicity of compatible cost-like objects (Baheri, 2023).

6. Misconceptions, limitations, and current directions

A recurring misconception is that an observed optimal plan determines a unique cost. In discrete entropic OT, this is false even in the noiseless case: the cost is identified only modulo additive row and column terms, or equivalently only through its cross-ratios (Chiu et al., 2021). In continuous smooth OT, local uniqueness reappears only after quotienting out the same natural invariances and after imposing regularity assumptions strong enough to generate curvature (Peyré et al., 24 Apr 2026).

Another misconception is that more regularization automatically improves inverse recovery. Entropic smoothing indeed makes the forward map strictly positive and differentiable, but inverse recovery can remain ill-conditioned. In sparse IOT, the sample complexity carries an $C$8 factor, so very small $C$9 is statistically harmful, while very large $\hat\pi$0 changes the effective model class by pushing the problem toward a Lasso-like regime (Andrade et al., 2023). For Bregman-regularized IOT, adding a small strongly convex penalty improves convergence of block coordinate methods, but numerical experiments show that the penalty must be extremely small to avoid biasing the recovered cost (Bao et al., 4 Oct 2025).

The sharpest divide in the field remains the one between discrete and smooth continuous settings. Discrete IOT reduces to inverse linear programming and is generically flat; continuous IOT can exhibit strict second-order structure, but this disappears again in highly symmetric regimes such as affine OT maps between Gaussian or elliptical marginals (Peyré et al., 24 Apr 2026). A plausible implication is that identifiability in practice depends as much on the geometry of the observed distributions as on the choice of inverse algorithm.

Open directions are therefore structural rather than merely computational. Current work points toward at least four fronts: richer nonparametric cost classes beyond entropy-based discrete models; systematic identifiability theory under multiple observed couplings or partial OT information; statistical theory for Bregman-regularized and nonlinear inverse formulations; and integration of OT-based inverse recovery with broader inverse-problem settings, including unbalanced transport and PDE-constrained models (González-Sanz et al., 2023, Bao et al., 4 Oct 2025, Lee et al., 20 May 2026). This suggests that IOT is evolving from a narrow “recover the cost from the plan” problem into a general theory of inferring latent transport structure from optimality data.