Decomposition Theorem for Optimal Transport

• The decomposition theorem for optimal transport plans is a framework that expresses complex plans as canonical mixtures over simpler, structured sub-plans.
• It underpins both theoretical insights and numerical methods by reducing high-dimensional problems into tractable, lower-dimensional cases.
• Different cost settings such as convex, concave, and martingale lead to distinct decomposition patterns, ensuring unique and efficient transport solutions.

The decomposition theorem for optimal transport plans refers generically to rigorous structural results that express the optimal transport plan (or the class of all optimal plans) as a canonical sum, disintegration, or mixture over simpler or more structured sub-plans. These results elucidate the internal architecture of optimal couplings under varying cost functions—including convex, concave, norm, and more general functionals—and often under specific structural constraints (martingale, invariance, etc). Decomposition theorems underpin both theoretical understanding and numerical approaches to high-dimensional and constrained transport by reducing complex instances to ensembles of lower-dimensional, uniquely structured, or tractable problems.

1. Decomposition in Classical and Invariant Optimal Transport

A foundational example is the ergodic/barycentric decomposition for invariant Kantorovich problems. Given a simplex of invariant probability measures $\Dom$ under a group action or transformation $T$ on a Polish space $X$, one has a unique barycentric (Choquet) decomposition: $\mu = \int_{\Ext(\Dom)} e\, d\alpha_\mu(e),\quad \nu = \int_{\Ext(\Dom)} f\, d\alpha_\nu(f)$ where $\Ext(\Dom)$ is the set of ergodic (extreme-point) measures. The decomposition theorem states that any invariant optimal transport plan $\pi^*$ admits the representation

$\pi^* = \int_{\Ext(\Dom)\times\Ext(\Dom)} \pi^*(e, f)\, d\rho(e, f)$

where $\rho$ is a coupling of the ergodic decompositions, and each $\pi^*(e, f)$ is an ergodic optimal plan for the pair $(e, f)$. The invariant Wasserstein metric similarly decomposes as a $p$-Wasserstein distance between barycentric measures on the extreme boundary, i.e.,

$W^G_p(\mu, \nu) = \inf_{\rho \in \Pi(\alpha_\mu, \alpha_\nu)} \left(\int_{\Ext(\Dom)\times\Ext(\Dom)} W^G_p(e, f)^p\, d\rho(e, f)\right)^{1/p}$

This structure allows the reduction of transport problems on the simplex to combinations of extremal problems, and is robust to linear constraints and further generalizations (Zaev, 2015).

2. Sudakov-Type and Affine Partition Decompositions for Convex Costs

For transport with convex cost functions $c:\mathbb{R}^d \to [0,\infty]$, decomposition theorems following Sudakov's principle partition the space into a family ${S^h_\mathfrak a}$ of locally affine sets of varying dimension, labeled by $(h, \mathfrak a)$, such that

• $$\mathbb{R}^d = \bigsqcup_{h, \mathfrak a} S^h_\mathfrak a$$ up to Lebesgue-null sets,
• On each $S^h_\mathfrak a$ the displacement $x' - x$ of the optimal transport is restricted to a proper, parallel $h$-dimensional face $O^h_\mathfrak a$ of $\text{epi}\,c$,
• Each $S^h_\mathfrak a$ is cyclically connected and indecomposable for the face-induced cost,
• The Lebesgue measure decomposes regularly across these pieces.

On each cell, Monge maps exist as secondary minimizers (e.g., for strictly convex costs on conditional problems), and the global optimal plan is reassembled from the local cell-wise solutions. This fine stratification significantly advances the classic theory by enabling existence and partial uniqueness statements for Monge-type solutions even in non-smooth or highly non-strictly convex cases (Bianchini et al., 2014, Bianchini et al., 2013).

3. Decomposition for Concave Costs and Singular Measures

When the cost is a strictly concave function of the distance $\ell(\|x - y\|)$, the decomposition theorem guarantees every optimal transport plan $\gamma$ takes the form: $$\gamma = (id, id)_{\#}(\mu \wedge \nu) + (id, T)_{\#}((\mu - \nu)^+)$$ where $(\mu \wedge \nu)$ is the largest common part (carried diagonally), $(\mu - \nu)^+$ and $(\nu - \mu)^+$ are the mutually singular remainders, and $T$ is a unique map pushing $(\mu - \nu)^+$ to $(\nu - \mu)^+$. All shared mass remains at rest on the diagonal (forced by strict subadditivity of the cost), and the transport between the singular parts is realized uniquely by a Monge map, provided the source gives no mass to $(d-1)$-rectifiable sets (Pegon et al., 2013). This splitting sharply delineates mass-fixation vs movement and extends classic results under minimal regularity.

4. Martingale and Generalized Monotone Decompositions

In martingale optimal transport (MOT), decomposition hinges on partitioning $\mathbb{R}^d$ (the source space) via a unique "convex paving" or irreducible paving: $x \mapsto I(x)$, with each $I(x)$ a non-empty convex cell. Any optimal martingale transport $\pi$ can be written as

$\pi = \int_{I(\mathbb{R}^d)} \pi_I\, \eta(dI)$

with each $\pi_I$ the restriction of $\pi$ to $I \times \mathbb{R}^d$, and the barycenters always in $I$. On every convex cell $I$, the restricted problem is itself an (irreducible) MOT problem, the duality is attained in a pointwise (non-quasi-sure) sense, and the conditional coupling $\pi_{x}$ is supported on the extreme points of its conditional convex hull. This structure enables local analysis of both primal and dual problems and yields full characterizations in dimensions $d\leq 3$ (and more generally under analytic set-theoretic hypotheses) (March, 2018, Ghoussoub et al., 2015).

5. One-Dimensional Decomposition of Optimal Plans

For $\mathbb{R}$ and cost $c(x, y) = |x - y|$, all optimal plans admit a canonical unique decomposition indexed by the monotonicity of the difference of the cumulative distribution functions. Writing

$$F^+_\mu(x) = \mu((-\infty, x]), \quad F^+_\nu(x) = \nu((-\infty, x])$$

define regions $E^+ = \{x : F^+_\mu(x) > F^+_\nu(x)\}$, $E^- = \{x : F^+_\mu(x) < F^+_\nu(x)\}$, $E^= = \{x : F^+_\mu(x) = F^+_\nu(x)\}$, with corresponding measures $\mu^+, \nu^+, \mu^-, \nu^-, \mu^=, \nu^=$. The decomposition theorem (atomless case) asserts

$$O(\mu, \nu) = \Bigl(\bigoplus_{k \in K^+} O(\mu^+_k, \nu^+_k)\Bigr) \oplus \Bigl(\bigoplus_{k \in K^-} O(\mu^-_k, \nu^-_k)\Bigr) \oplus O(\mu^=, \nu^=)$$

Every optimal plan decomposes uniquely as a sum of plans on these non-overlapping intervals, each component addressing monotone, anti-monotone, or "fixed" mass (i.e., where the CDFs agree). This makes explicit the connection between cyclic monotonicity, CDF "barrier points," and decomposition of transport behavior into forward, backward, and identity types (Ley, 4 Dec 2025).

6. Applications, Variants, and Algorithmic Implications

Decomposition theorems underlie both theoretical and algorithmic progress:

• In invariant transport, they yield metric reconstruction for Wasserstein distances between invariant (e.g., stationary or ergodic) measures and handle transport under further constraints by lifting to the boundary/extremal structure (Zaev, 2015).
• The Sudakov and affine-stratification decompositions are central to proving existence and (essential) uniqueness of Monge maps, reducing high-dimensional problems to cell-wise low-dimensional instances with tractable geometries (Bianchini et al., 2014, Bianchini et al., 2013).
• In martingale and barycentric cost settings, the structure clarifies the role of convex order and quantifies splitting into deterministic and stochastic martingale parts (as in the mixture of Brenier and Strassen theorem) (Gozlan et al., 2018).
• For line transport with distance cost, decomposition enables rigorous analysis of entropic regularizations and their zero-entropy limits, clarifying the emergence and uniqueness of limiting plans (Ley, 4 Dec 2025).

7. Comparative Table of Major Decomposition Theorems

Setting / Cost	Form of Decomposition	Canonical Plan Structure
Invariant OT, general cost	Mixture over boundary (ergodic) plans	$\pi^* = \int \pi^*(e, f)\, d\rho(e,f)$
Convex cost in $\mathbb{R}^d$	Partition into affine/cyclically connected cells	Piecewise Monge maps on each cell
Strictly concave cost	Diagonal (fixed) + Monge map on singular remainders	$$\gamma = (id,id)_\#(\mu\wedge\nu) + (id,T)_\#((\mu-\nu)^+)$$
Martingale OT in $\mathbb{R}^d$	Disintegration over irreducible convex pavings	Each cell supports extremal-splitting martingale
$|x-y|$ cost on $\mathbb{R}$	Sum over positive, negative, frozen (barrier) regions	Direct sum of cyclically monotone subplans

Each variant entails a precise mechanism for splitting the transport task into canonical, lower-dimensional, or more structured problems, effectively reducing complexity and revealing fine geometric regularity of optimal couplings.

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References:

(Zaev, 2015, Bianchini et al., 2014, Pegon et al., 2013, Ghoussoub et al., 2015, March, 2018, Gozlan et al., 2018, Bianchini et al., 2013, Ley, 4 Dec 2025)