Wasserstein Formulation & Optimal Transport

• Wasserstein formulation is a framework in optimal transport that minimizes the cost of moving probability measures, typically using squared Euclidean distances.
• It extends to multi-marginal and dual settings with strategies like genetic column updates to enable sparse, tractable solutions for high-dimensional problems.
• Applications include computing barycenters, mesh-free interpolation, and Wasserstein splines, advancing techniques in statistics, machine learning, and computational mathematics.

The Wasserstein formulation is a canonical framework in optimal transport theory that characterizes the minimal cost of transporting one probability measure to another with respect to a specified ground cost, most commonly the squared Euclidean distance. This formulation encompasses both classical two-marginal problems and generalizations to the multi-marginal setting, where optimal transport plans couple more than two marginals. Modern developments include dual formulations, algorithmic advances for tractable high-dimensional computation, as well as applications to Wasserstein barycenters and spline interpolation in the metric space of probability measures. The Wasserstein framework enables both geometric and variational perspectives, which are foundational for recent advances in statistics, machine learning, computational mathematics, and applied sciences (Friesecke et al., 2022).

1. Multi-Marginal Optimal Transport: Primal and Dual Wasserstein Formulations

Let $X\subset\mathbb{R}^d$ and given probability measures $\mu_k\in\mathcal{P}(X)$ for $k=1,\dots,N$, the multi-marginal Kantorovich problem is

$$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$

where $\Pi(\mu_1,\dots,\mu_N)$ denotes the set of all couplings with prescribed marginals

$$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$

and $c:X^N\rightarrow\mathbb{R}$ is a given cost function. For quadratic cost, the Wasserstein-$2$ setting, $c(x_1,\ldots,x_N)$ is typically a sum of squared distances or a symmetric quadratic cost.

After discretization, each marginal is represented on a finite grid $X_k=\{a^{(k)}_1,\dots,a^{(k)}_{\ell_k}\}$, and a plan $\mu_k\in\mathcal{P}(X)$0 is an $\mu_k\in\mathcal{P}(X)$1 nonnegative tensor, interpreted as a probability mass function. The discrete problem reads

$\mu_k\in\mathcal{P}(X)$2

where $\mu_k\in\mathcal{P}(X)$3 extracts the $\mu_k\in\mathcal{P}(X)$4-th marginal, and $\mu_k\in\mathcal{P}(X)$5 is the Frobenius inner product. The feasible set is the so-called Kantorovich polytope.

The dual formulation introduces potentials $\mu_k\in\mathcal{P}(X)$6, yielding

$\mu_k\in\mathcal{P}(X)$7

with equality of primal and dual optima by linear programming duality.

2. The GenCol Algorithm for High-Dimensional Multi-Marginal Problems

The GenCol (genetic column generation) algorithm is specifically designed to solve large-scale multi-marginal optimal transport (MMOT) linear programs which arise in mesh-based Wasserstein formulations. The algorithm operates on a restricted subset of the configuration space, iteratively augmenting active constraints to converge to the full optimal solution. The main steps are:

1. Column restriction: Start with a small subset $\mu_k\in\mathcal{P}(X)$8.
2. Reduced LP solve: Solve the primal and dual LPs on $\mu_k\in\mathcal{P}(X)$9.
3. Feasibility check: If dual feasibility extends globally, terminate; the current solution is optimal.
4. Genetic update: Otherwise, select a "parent" $k=1,\dots,N$0 with $k=1,\dots,N$1, mutate one coordinate to create an "offspring" $k=1,\dots,N$2 not in $k=1,\dots,N$3, and admit $k=1,\dots,N$4 if it violates the dual constraint.
5. Pruning: If $k=1,\dots,N$5 exceeds a chosen inflation threshold, prune inactive columns.

Key properties include:

• Primal LP on $k=1,\dots,N$6 has $k=1,\dots,N$7 variables and $k=1,\dots,N$8 constraints.
• By sparsity, at most $k=1,\dots,N$9 columns are needed for an exact solution.
• The genetic update ensures efficient exploration while controlling computational complexity.

The GenCol algorithm empirically achieves exponential convergence and enables MMOT problems with $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$0–$$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$1 variables to be solved on standard hardware, whereas classical approaches (Sinkhorn iterations, iterative Bregman projections) are infeasible due to memory constraints (Friesecke et al., 2022).

3. Wasserstein Barycenters and Mesh-Free Formulations

The Wasserstein barycenter is the probability measure that minimizes a weighted sum of Wasserstein distances to a given family of marginals. The standard barycenter problem is

$$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$2

with $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$3, $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$4. In the mesh-free multi-marginal formulation, the cost is

$$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$5

with $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$6. The barycenter measure is then the pushforward $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$7, where $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$8 is the optimal MMOT plan, allowing the barycenter to be supported on a grid of size $$\min_{\gamma\in\Pi(\mu_1,\dots,\mu_N)} \int_{X^N} c(x_1,\ldots,x_N)\;\mathrm d\gamma(x_1,\ldots,x_N),$$9 times finer than the input marginals—hence, "mesh-free."

This formulation avoids limitations of grid-based approaches, which restrict barycenter locations, by exploiting the higher-dimensional simplex induced by the multi-marginal coupling (Friesecke et al., 2022).

4. Wasserstein Splines and Geodesic Interpolation

Wasserstein splines generalize classical spline interpolation to the space of probability measures endowed with the Wasserstein metric. Given marginal constraints at discrete times $\Pi(\mu_1,\dots,\mu_N)$0, the MMOT problem seeks a joint law $\Pi(\mu_1,\dots,\mu_N)$1 on $\Pi(\mu_1,\dots,\mu_N)$2 minimizing

$\Pi(\mu_1,\dots,\mu_N)$3

subject to $\Pi(\mu_1,\dots,\mu_N)$4 for all $\Pi(\mu_1,\dots,\mu_N)$5. The cost $\Pi(\mu_1,\dots,\mu_N)$6 is defined via

$\Pi(\mu_1,\dots,\mu_N)$7

the cubic-spline energy associated with the knot sequence. The resulting interpolants are $\Pi(\mu_1,\dots,\mu_N)$8, where $\Pi(\mu_1,\dots,\mu_N)$9 evaluates the spline at intermediate times.

This provides a principled, variational approach to interpolate measures along smooth paths in Wasserstein space, with links to dynamic optimal transport (Friesecke et al., 2022).

5. Sparsity, Correctness, and Complexity of Wasserstein Formulations

Several theoretical properties characterize the tractability of the MMOT Wasserstein formulation:

• Sparsity (Dubins): Any optimizer $$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$0 to the discrete MMOT LP with marginal grid sizes $$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$1 has at most $$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$2 nonzeros. This sharp bound ensures that although the full coupling grows exponentially, optimizers are supported on a much smaller subset of configurations.
• Algorithmic correctness (GenCol): Under the genetic column update, the reduced LP converges to the global optimum once dual feasibility is satisfied, ensuring correctness and finite termination in practice.
• Computational efficiency: Each iteration involves LPs of size $$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$3, with overall complexity dictated by the manageable active set size $$\Pi(\mu_1,\dots,\mu_N) =\left\{\gamma\in\mathcal{P}(X^N)\;\left|\; (\pi_k)_\#\gamma=\mu_k\quad\forall\,k=1,\dots,N\right.\right\},$$4.

Empirical observations confirm rapid (often exponential) convergence and mesh-free numerical accuracy for barycenter and spline tasks in high dimensions (Friesecke et al., 2022).

6. Significance: Applications and Extensions

The general Wasserstein formulation underlies a broad range of optimal transport-based techniques. Key applications include:

• Multi-marginal barycenters: Allowing mesh-free, high-dimensional computation, essential for statistical averaging in geometry and machine learning.
• Wasserstein splines: Enabling smooth interpolation of distributions in generative modeling, time-series, and dynamical inference.
• High-dimensional OT solvers: Via the GenCol scheme and sparsity guarantees, practical in scientific computing regimes previously inaccessible to full dense plan approaches.
• Theoretical innovation: The integration of duality, sparsity, and mesh-free evaluation within a unified mathematical framework has enabled advances across computational mathematics, statistics, and applied sciences.

References and further technical details, such as precise convergence proofs or spline cost expressions, are detailed in (Friesecke et al., 2022).