Optimal Transport Independence Criterion

• The topic defines independence as the optimal transport cost between the joint distribution and the product of its marginals, with a cost of zero indicating independence.
• It includes various formulations—exact Wasserstein, entropic Sinkhorn, and algebraic approaches—each offering unique computational and testing insights.
• These criteria support metric-aware, nonparametric tests for independence, with applications in censored data analysis, fairness, and invariant feature learning.

Searching arXiv for recent and foundational papers on optimal transport-based independence criteria and related formulations. {"query":"optimal transport independence criterion entropic optimal transport independence criterion transport dependency rank joint distance covariance optHSIC arXiv", "max_results": 10} Optimal transport-based independence criterion denotes a family of dependence measures and test procedures that quantify departure from independence by comparing a joint law to an independence-constrained reference through an optimal transport objective. In the most direct formulation, one measures the transport cost between the joint distribution and the product of its marginals; in other formulations, independence is enforced through transport to an algebraic independence variety, through transport to a common barycenter that removes nuisance or censoring effects, or through rank maps induced by optimal transport (Nies et al., 2021, Çelik et al., 2019, Lipnick et al., 4 Jul 2025, Niu et al., 2022). Across these variants, the organizing idea is the same: independence is encoded as a structural constraint on a target distribution or representation, and optimal transport supplies a geometry-aware discrepancy or transformation mechanism. This yields criteria that are metric-aware, often nonparametric, and in several settings admit exact independence characterizations, calibrated testing procedures, or explicit optimization structures (Nies et al., 2021, Liu et al., 2021).

1. Foundational definitions and scope

An optimal transport-based independence criterion is most explicitly defined by comparing the joint distribution to the product of the marginals. For random variables $X$ and $Y$, the “transport dependency” paper defines

$$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$

where $\gamma$ is the joint law, $p^X_\#\gamma$ and $p^Y_\#\gamma$ are its marginals, and $T_c$ is the Kantorovich optimal transport cost under a positive cost $c$ on $X \times Y$ (Nies et al., 2021). In the same source, if $c$ is positive, then

$Y$0

so vanishing transport dependency is equivalent to independence (Nies et al., 2021).

A discrete independence criterion of the same type appears in Wasserstein-distance-based testing, where one uses the empirical Wasserstein distance between the empirical joint $Y$1 and the empirical product measure $Y$2 (Xie et al., 2022). The paper formulates the statistic as

$Y$3

with an additive ground metric on the product space, and states that $Y$4 (Xie et al., 2022).

A second major line replaces exact OT by entropy-regularized OT. The “Entropy Regularized Optimal Transport Independence Criterion” defines the population criterion as the Sinkhorn divergence between the joint and the product of marginals,

$Y$5

and states that if the Gibbs kernels are positive-universal then $Y$6 and $Y$7 iff $Y$8 (Liu et al., 2021).

A third line treats independence as a geometric model class rather than a product target. In “Optimal Transport to a Variety,” the independence model $Y$9 is the set of product distributions on a finite product space, and the criterion is

$$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$0

Independence holds iff $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$1 (Çelik et al., 2019). This formulation turns the independence problem into “optimal transport to a variety” (Çelik et al., 2019).

These formulations are not interchangeable in a literal sense. A plausible implication is that “optimal transport-based independence criterion” is best regarded as a class of related constructions rather than a single canonical statistic.

2. Main mathematical formulations

The direct Wasserstein formulation begins from the discrete or continuous Kantorovich problem. In the transport dependency framework, for a lower semicontinuous symmetric cost $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$2,

$$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$3

and the independence criterion is the OT cost from the joint to the product law (Nies et al., 2021). This construction is defined on general Polish spaces and “intrinsically respects metric properties” (Nies et al., 2021). The same paper also introduces normalized transport correlations $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$4, $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$5, and $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$6, each taking values in $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$7, with extremal characterizations in terms of $$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$8-Lipschitz maps, measurable functional dependence, or dilatations (Nies et al., 2021).

The entropic variant replaces exact OT by

$$\tau(\gamma) = \tau_c(\gamma) = T_c\!\big(\gamma,\, p^X_\#\gamma \otimes p^Y_\#\gamma\big),$$9

and then debiases it through the Sinkhorn divergence

$\gamma$0

(Liu et al., 2021). The resulting ETIC statistic inherits sensitivity to both linear and nonlinear dependence and admits non-asymptotic bounds under sub-Gaussian or bounded-support assumptions (Liu et al., 2021).

The algebraic-geometry formulation minimizes Wasserstein distance to the independence variety. For $\gamma$1, the independence model is the rank-one locus of the probability matrix, and in the two-bit case it is characterized by

$\gamma$2

(Çelik et al., 2019). The induced objective $\gamma$3 becomes a minimum over a union of transportation polytopes, piecewise linear or quadratic on cells determined by the ground metric (Çelik et al., 2019).

Several papers replace “distance to independence” by “transport to independence.” In the Monge optimal transport barycenter problem, independence is encoded by requiring a transformed outcome $\gamma$4 to satisfy $\gamma$5, which is equivalent to pushing each conditional law $\gamma$6 to a common barycenter $\gamma$7 (Lipnick et al., 4 Jul 2025). The fairness-oriented sigma-algebra framework similarly defines

$\gamma$8

and shows that the minimizer is obtained by transporting each conditional law $\gamma$9 to a common Wasserstein barycenter $p^X_\#\gamma$0 (Bhattacharya et al., 9 Oct 2025). In the Gaussian conditional-independence setting, the nuisance variable $p^X_\#\gamma$1 is replaced by a barycentric residual $p^X_\#\gamma$2, and one penalizes $p^X_\#\gamma$3 as a proxy for $p^X_\#\gamma$4 (Bounos et al., 24 Dec 2025).

3. Statistical testing, calibration, and consistency

Optimal transport-based independence criteria are used both as descriptive dependence measures and as formal test statistics. ETIC constructs the empirical test statistic

$p^X_\#\gamma$5

and calibrates it by permutation (Liu et al., 2021). The paper proves non-asymptotic expectation and tail bounds, states that under $p^X_\#\gamma$6, $p^X_\#\gamma$7, and concludes that critical values scale as $p^X_\#\gamma$8 (Liu et al., 2021). Under fixed alternatives with $p^X_\#\gamma$9, power tends to $p^Y_\#\gamma$0 (Liu et al., 2021).

In Wasserstein-distance testing with empirical joint versus empirical product, permutation is also the natural calibration device. The testing pipeline computes $p^Y_\#\gamma$1, permutes $p^Y_\#\gamma$2 relative to $p^Y_\#\gamma$3, recomputes the statistic, and forms an empirical $p^Y_\#\gamma$4-value (Xie et al., 2022). The paper presents this as a consistent test under broad conditions studied in the literature, while focusing its own contribution on computation (Xie et al., 2022).

A distinct route to testing uses OT-based multivariate ranks. “Distribution-free joint independence testing and robust independent component analysis using optimal transport” defines OT rank maps by transporting each marginal to a fixed reference law and then constructs rank joint distance covariance, RJdCov, as a higher-order rank analogue of distance covariance (Niu et al., 2022). Under the null, the joint distribution of empirical ranks is uniform over permutations of the reference grid, so the resulting test is exactly distribution-free in finite samples and does not require moment assumptions because the rank distances are bounded on $p^Y_\#\gamma$5 (Niu et al., 2022). The paper states that RJdCov is zero iff the variables are mutually independent, that the sample estimator is strongly consistent, and that the test is universally consistent with nontrivial Pitman efficiency against contiguous alternatives (Niu et al., 2022).

In censored survival analysis, standard permutation arguments break down because one observes $p^Y_\#\gamma$6 rather than the lifetime $p^Y_\#\gamma$7. The optHSIC procedure addresses this by using optimal transport to transform the censored dataset into an uncensored one “while preserving the relevant dependencies,” then applying a kernel independence test to the transported data (Rindt et al., 2019). The abstract states that “The type 1 error is proven to be correct in the case where censoring is independent of the covariate” and that experiments indicate broader power than Cox proportional hazards regression (Rindt et al., 2019). This suggests that OT can be used not only to define a discrepancy but also to restore a valid testing geometry when the observed sample is structurally distorted.

4. Computational structure and algorithms

The computational burden of OT-based independence is a central issue, and the literature develops markedly different algorithmic responses.

For entropy-regularized criteria, Sinkhorn scaling is the basic solver. ETIC exploits the fact that under additive costs on Cartesian-product supports the Gibbs kernel factorizes as $p^Y_\#\gamma$8, allowing “Tensor Sinkhorn” updates

$p^Y_\#\gamma$9

with $T_c$0 time and $T_c$1 space per iteration rather than the naive $T_c$2 or $T_c$3 storage associated with an explicit $T_c$4 kernel (Liu et al., 2021). The same paper also offers a random-feature approximation to reduce per-iteration complexity to $T_c$5 for approximate Gibbs kernels (Liu et al., 2021).

For exact Wasserstein testing against the empirical product measure, the discrete OT instance has a one-to-many assignment structure: $T_c$6 has $T_c$7 atoms of mass $T_c$8, while $T_c$9 has $c$0 atoms of mass $c$1 (Xie et al., 2022). “Solving a Special Type of Optimal Transport Problem by a Modified Hungarian Algorithm” rewrites this as a semi-assignment problem with column capacities and proves that its modified Hungarian algorithm solves the resulting OT exactly in $c$2, which becomes $c$3 in the independence-test case $c$4, improving on the $c$5 order of classical Hungarian reduction (Xie et al., 2022).

For OT to an independence variety, the optimization is piecewise algebraic. The cost becomes a linear functional over a union of transportation polytopes indexed by simplices in a regular triangulation determined by the ground metric (Çelik et al., 2019). In the two-bit case there are $c$6 maximal simplices for the Hamming metric, and each piece yields an affine or quadratic objective in the product parameters $c$7, with explicit candidate optima listed in tables (Çelik et al., 2019).

For OT-rank-based independence, the main computational step is the empirical OT rank map. In the exact form this is an assignment problem of complexity $c$8, though entropic OT and geometric approximations are proposed as scalable alternatives (Niu et al., 2022).

The negative side is also sharp. “Discrete Optimal Transport with Independent Marginals is #P-Hard” proves that computing the Wasserstein distance when one distribution has independent discrete components is #P-hard even when the other distribution has only two atoms, and even for approximate solutions of specified additive accuracy (Taşkesen et al., 2022). Since the direct criterion $c$9 is exactly of this form, the paper implies that generic exact computation of OT-based independence is intractable in the worst case (Taşkesen et al., 2022). The same source develops a pseudo-polynomial dynamic-programming approximation for special lattice-structured cases and strongly polynomial algorithms for special Bernoulli-support instances (Taşkesen et al., 2022).

5. Variants tailored to structure: censoring, conditional independence, fairness, and invariance

Optimal transport-based independence criteria have been adapted to several structured statistical problems in which the dependence target is not plain unconditional independence.

In right-censored survival analysis, optHSIC combines OT with HSIC. The data are $X \times Y$0 with $X \times Y$1 and $X \times Y$2. The procedure transports the censoring-distorted time distribution to an estimate of the uncensored lifetime distribution without using $X \times Y$3, yielding a surrogate $X \times Y$4 on which permutation-based HSIC can be applied (Rindt et al., 2019). The provided technical exposition states that under independent censoring and $X \times Y$5, the transported surrogate remains asymptotically independent of $X \times Y$6, so permutation $X \times Y$7-values control type I error (Rindt et al., 2019).

For conditional independence and data repair, OTClean defines an OT-based CI discrepancy

$X \times Y$8

in the discrete case, or the corresponding integral over $X \times Y$9 in the continuous case (Pirhadi et al., 2024). The framework transports the empirical distribution to a repaired distribution that satisfies $c$0, balancing transport cost against CI enforcement through either hard quadratic constraints or regularized Sinkhorn-style updates (Pirhadi et al., 2024).

For certifiably fair representation learning, the sigma-algebra formulation compiles sensitive attributes and proxies into a sub-$c$1-algebra $c$2, then solves

$c$3

The optimal representation is obtained by transporting each conditional law $c$4 to a common Wasserstein barycenter $c$5, thereby enforcing distributional independence from $c$6 rather than mere decorrelation (Bhattacharya et al., 9 Oct 2025).

For invariant feature extraction, the Gaussian OTBP paper uses the Monge optimal transport barycenter problem for the nuisance $c$7. It defines $c$8 so that $c$9, and in the jointly Gaussian case obtains the explicit residual form

$Y$00

The key theorem states

$Y$01

for linear $Y$02 under the stated Gaussian assumptions (Bounos et al., 24 Dec 2025). This makes OT-based independence an implementable proxy for conditional invariance rather than a generic distance between distributions.

A related but more general barycentric view appears in the Monge OT barycenter paper, where independence $Y$03 is itself the pushforward constraint defining the barycenter problem (Lipnick et al., 4 Jul 2025). There, independence is relaxed to adversarial uncorrelation over finite-dimensional test spaces, and the adversary reduces to singular values of a small cross-feature matrix (Lipnick et al., 4 Jul 2025). This is not a hypothesis test in the classical sense, but it uses independence as the organizing OT constraint.

6. Relations to kernels, distance covariance, and common misconceptions

A recurrent misconception is that OT-based independence is simply another name for HSIC or distance covariance. The literature shows overlap but not equivalence.

ETIC is explicitly positioned against HSIC: with additive costs and $Y$04, the entropic criterion tends to $Y$05, whereas with multiplicative costs it tends to $Y$06 (Liu et al., 2021). This identifies a precise asymptotic connection without collapsing the methods into one another. The Monge barycenter formulation states that if the test spaces $Y$07 and $Y$08 are chosen as RKHS feature maps, the adversarial cross-covariance “reduces to maximizing the leading singular value of a kernel cross-covariance, akin to HSIC,” while remaining computationally distinct because it works with small $Y$09 matrices and per-sample gradients (Lipnick et al., 4 Jul 2025).

Another misconception is that OT-based criteria are always purely nonparametric distances from the joint to the product law. The algebraic variety approach instead measures distance to the set of independent models under a chosen ground metric (Çelik et al., 2019). The fairness and invariance constructions solve constrained barycenter problems where independence is a property of the transported representation rather than of the original variables (Bhattacharya et al., 9 Oct 2025, Bounos et al., 24 Dec 2025). RJdCov uses OT only to construct multivariate ranks, after which the test statistic itself is a rank-distance covariance (Niu et al., 2022). The phrase therefore spans discrepancy-based, transformation-based, and rank-induced procedures.

A further misconception is that OT automatically guarantees tractable computation. The #P-hardness result for discrete OT with independent marginals shows the opposite in worst-case settings (Taşkesen et al., 2022). Exactness is sometimes feasible only because of special structure, as in the modified Hungarian algorithm for empirical product-versus-joint testing (Xie et al., 2022), tensorized Sinkhorn for entropic OT on Cartesian products (Liu et al., 2021), or two-bit algebraic decomposition for transport to an independence variety (Çelik et al., 2019).

Finally, independence versus decorrelation is treated explicitly in several papers. The fairness sigma-algebra framework states that decorrelating $Y$10 from $Y$11 is insufficient because it only aligns first moments, whereas OT equalizes the full conditional law across $Y$12-fibers (Bhattacharya et al., 9 Oct 2025). The Gaussian OTBP work similarly distinguishes $Y$13 from weaker covariance conditions, though in the jointly Gaussian case the two coincide because zero covariance is equivalent to independence (Bounos et al., 24 Dec 2025). This suggests that OT-based criteria are often adopted precisely when full distributional invariance, not moment-level neutrality, is the target.

Optimal transport-based independence criteria therefore constitute a broad research program rather than a single formula. Their shared contribution is to encode independence through transport geometry: as Wasserstein distance to a product law, Sinkhorn divergence to independence, Wasserstein distance to an independence variety, an OT rank map yielding a distribution-free statistic, or a barycentric transformation that removes nuisance structure while preserving utility (Nies et al., 2021, Liu et al., 2021, Çelik et al., 2019, Niu et al., 2022, Lipnick et al., 4 Jul 2025). The choice among these forms depends less on abstract generality than on the scientific object of independence—unconditional dependence, censored survival dependence, conditional independence, fairness with respect to a $Y$14-algebra, or invariant representation learning.