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Quadratic-form Optimal Transport

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Pith's one-line read This paper introduces quadratic-form optimal transport, a bilinear analogue of Kantorovich transport, and shows that for rectangular and related costs the unique minimizer is the diamond transport, a coupling supported on a diamond in the…

desk verdict The diamond transport and the QOT framework are real contributions; Theorem 6.2 is solid, but Theorem 6.9's moment assumption doesn't support the stability step. read the letter →

arxiv 2501.04658 v5 pith:PK22UZC5 submitted 2025-01-08 math.PR math.OC

Ruodu Wang , Zhenyuan Zhang This is my paper · ORCID
classification math.PRmath.OC MSC 49Q2262H0591B7062H20
keywords quadratic-formoptimaltransportdiamondcopulaGromov-WassersteindistancequadraticassignmentproblemsubmodularitycompletelymonotonefunctionsKantorovich
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quadratic-form optimal transport (QOT) replaces the linear cost $\int c\,d\pi$ of classical optimal transport with the bilinear cost $\int\!\!\int c\,d\pi\otimes d\pi$, so the planner pays the expected cost between two independent draws from the same coupling. The paper's central claim is that, despite the non-convexity this creates, several natural cost classes have explicit minimizers, the most striking being the diamond transport $\pi_{\mathrm{dia}}$, a coupling whose copula is uniform on the diamond $\{|u-1/2|+|v-1/2|=1/2\}$. For the rectangular cost $c(x,y,x',y')=|(x-x')(y-y')|$, $\pi_{\mathrm{dia}}$ is the unique minimizer for any marginals with finite first moments, which solves the inequality-minimization problem in the introduction. For products of completely monotone functions and for $|(x-x')(y-y')|^q$ with $q\in(1,2]$, $\pi_{\mathrm{dia}}$ minimizes whenever both marginals are symmetric. The paper also solves other QOT classes with comonotone, antimonotone, mixed, and V-shaped couplings, connecting the framework to Gromov-Wasserstein distances, quadratic assignment problems, Kendall's tau, covariance, and quadratically regularized optimal transport.

What carries the argument

The central object is the diamond copula $C_{\mathrm{dia}}$, the cdf of the uniform distribution on $D=\{(u,v)\in[0,1]^2:|u-1/2|+|v-1/2|=1/2\}$, and the induced diamond transport $\pi_{\mathrm{dia}}=(Q_\mu(U),Q_\nu(V))$ with $(U,V)\sim C_{\mathrm{dia}}$. For the rectangular cost, the proof exploits the identity $|(x-x')(y-y')|=\int\!\!\int \mathbf{1}_{\{(u,v)\in[(x,y),(x',y')]\}}\,du\,dv$; for fixed $(u,v)$ the inner probability is a quadratic function of $A(u,v)=\pi((-\infty,u]\times(-\infty,v])$, and its unique minimizer over the feasible interval is exactly $C_{\mathrm{dia}}(F_\mu(u),F_\nu(v))$. For the completely monotone family, Schoenberg's theorem makes $\phi((x-x')^2)$ a positive definite kernel, so the QOT objective becomes convex by the Schur product theorem; a technical lemma then shows the averaged cost $\tilde c(x,y)=\int c(x,y,x',y')\,d\pi_{\mathrm{dia}}(x',y')$ is supermodular on the first and third quadrants and submodular on the second and fourth, forcing the symmetrized minimizer to coincide with $\pi_{\mathrm{dia}}$. For the $q$-rectangular costs the proof approximates $|x-x'|^q+|y-y'|^q$ by $\alpha^{-2}(e^{-\alpha(|x-x'|^q+|y-y'|^q)}-1+\alpha(|x-x'|^q+|y-y'|^q))$, applies the exponential case, and passes to the limit using QOT stability.

What would settle it

Discretize $\mu=\nu$ as the uniform distribution on $\{0,1/2,1\}$ and solve the 3 by 3 quadratic program (A.1) for the rectangular cost $c(x,y,x',y')=|(x-x')(y-y')|$. Theorem 6.2 predicts the unique minimizer is the diamond transport; any feasible coupling with strictly smaller cost than the discretized diamond transport would refute the theorem. The same experiment with asymmetric marginals and $q=2$ maps the boundary of Theorem 6.9.

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Extended reading notes

Core claim

The paper establishes that the QOT problem---minimize $\int\!\!\int c(x,y,x',y')\,d\pi(x,y)\,d\pi(x',y')$ over couplings $\pi\in\Pi(\mu,\nu)$---is a genuinely new optimization structure, not a variant of classical transport: it is generally non-convex, duality is not generally available, and optimizers need not be Monge maps. Its main positive discovery is that the diamond transport $\pi_{\mathrm{dia}}$ is a universal optimizer for several type-XX cost families. Theorem 6.2 gives the sharpest statement: for $c(x,y,x',y')=|(x-x')(y-y')|$ and $\mu,\nu\in P_1(\mathbb{R})$, $\pi_{\mathrm{dia}}$ is the unique minimizer; the proof writes the cost as the area of the rectangle spanned by the two points and minimizes a quadratic function of the coupling's cdf pointwise. Theorem 6.5 extends this to $c=\phi((x-x')^2)\phi((y-y')^2)$ with $\phi$ completely monotone and $\phi'(u)+2u\phi''(u)\le 0$, for symmetric marginals, using positive-definite-kernel convexity plus a quadrant-by-quadrant supermodularity lemma; Theorem 6.9 extends it to $|(x-x')(y-y')|^q$, $q\in(1,2]$, again under symmetry, by a limiting argument from the exponential case. Along the way the paper shows comonotone, antimonotone, X-shaped, and V-shaped couplings solve other explicit QOT classes, and it formulates the framework so that the Gromov-Wasserstein distance and the Koopmans-Beckmann quadratic assignment problem appear as special cases.

Load-bearing premise

For the broad non-rectangular diamond-transport theorems, both marginals must be symmetric about a common point, and for the completely monotone family the inequality $\phi'(u)+2u\phi''(u)\le 0$ is also load-bearing; without these, the paper does not claim the diamond transport minimizes.

Editorial extensions

If this is right

  • The inequality-minimization problem from the introduction is solved in closed form: the diamond transport minimizes the average squared weighted discrepancy $(\theta_1|X-X'|+\theta_2|Y-Y'|)^2$ between two randomly selected individuals, and for uniform marginals it gives every wealth level the same expected benefit.
  • The paper explicitly characterizes the maximizers of the $(2,1)$-Gromov-Wasserstein transport cost on the real line for arbitrary marginals with finite first moments, and the $(2,q)$-GW maximizers for $q\in(1,2]$ under symmetric marginals: in both cases the diamond transport attains them.
  • QOT optimizers are genuinely non-Monge: the Bernoulli example shows the independent coupling can be the unique minimizer, and the diamond transport is supported on a diamond rather than on a graph, so the Monge assumption used in quadratic assignment problems cannot be relaxed without changing the answer.
  • For several other cost classes---quadratic products, jointly submodular costs, Gromov-Wasserstein-type costs, and the separable costs of Theorem 5.10---the paper gives explicit minimizers that are comonotone, antimonotone, X-shaped, or V-shaped, providing a complete reference table of solvable QOT problems.
  • Since QOT contains the Koopmans-Beckmann quadratic assignment problem, the explicit diamond solution offers a benchmark for discrete QAP heuristics and a target for Monge approximations by permutation maps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves the completely monotone family without symmetry open; a natural next test is whether a four-piece 'diamond-type' coupling built from two comonotone and two antimonotone pieces minimizes that family, a shape the paper's Appendix D already conjectures.
  • The pointwise cdf-minimization method behind Theorem 6.2 may also work for other costs that factor as products of one-dimensional increments, such as $\min\{|x-x'|,|y-y'|\}$, which the paper lists as open.
  • If the diamond transport is the right allocation rule for inequality minimization, it predicts a specific testable pattern: conditional on wealth, the assigned benefit has constant mean, a property unlike comonotone or antimonotone matching.
  • The explicit $(2,q)$-GW maximizers on the real line could seed closed-form Gromov-Wasserstein solutions on discrete structures whose distance matrices embed into the line, though the paper does not establish that transfer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a new framework, quadratic-form optimal transport (QOT), where the cost is linear in the product measure π⊗π rather than in π. It develops general existence, stability, lower-bound, and convexity results, and it presents several explicit solution families. The central contribution is the diamond transport π_dia, a coupling whose copula is supported on a diamond, and the proof that it uniquely minimizes the rectangular cost |(x−x′)(y−y′)| for arbitrary one-dimensional marginals in P1 (Theorem 6.2). Further chapters extend diamond optimality to completely monotone costs under symmetric marginals (Theorem 6.5) and to q-rectangular costs for q∈(1,2] (Theorem 6.9), with additional results on V-transports, Gromov–Wasserstein-type costs, and quadratic assignment problems.

Significance. If the main claims are correct, this is a valuable systematic theory of a non-convex transport problem that includes inequality minimization, covariance-type objectives, Gromov–Wasserstein distances, and QAP as special cases. The proof of Theorem 6.2 is elegant and genuinely self-contained: it reduces the problem to pointwise minimization of a convex quadratic function of the coupling cdf A(u,v), and the diamond copula emerges directly from the clamp of the unconstrained minimizer to the feasible interval. The paper is also commendable for giving independent lower bounds, explicit couplings, and no fitted parameters. However, the advertised universality of the diamond coupling is narrower than the abstract suggests: the q>1 results require symmetric marginals, and one headline theorem (Theorem 6.9) has a load-bearing moment-assumption gap; the general stability proposition used there also has a proof gap.

major comments (2)
  1. [Theorem 6.9, verification of (4.1)] The statement assumes µ,ν∈P_{2+δ}(R), but the proof verifies the uniform integrability condition (4.1) by bounding E[|X−X′|^{q+δ/2}|Y−Y′|^{q+δ/2}] by E[|X−X′|^{2q+δ}]^{1/2}E[|Y−Y′|^{2q+δ}]^{1/2}. Since q>1, the exponent 2q+δ is strictly larger than 2+δ, so finiteness of the (2+δ)-th moments does not imply finiteness of the moments used in the bound. The displayed constant C(p,δ) cannot repair this, and for heavy-tailed marginals with finite 2+δ moments but no finite 2q moments the given Cauchy–Schwarz argument simply does not establish (4.1). Thus Theorem 6.9 is unproved as stated. The fix may be straightforward — strengthening the assumption to µ,ν∈P_{2q+δ}(R), or supplying a different approximation argument that avoids the uniform-in-π bound — but as written this is a concrete gap in a headline result.
  2. [Proposition 4.5, proof of cost convergence] The uniform integrability condition (4.1) is imposed only as a supremum over couplings of the limiting marginals µ,ν. In the proof, the assertion that ∫∫ c dπ_n⊗dπ_n → ∫∫ c dπ⊗dπ (following Van der Vaart [2000, Theorem 2.20]) requires uniform integrability of c along the approximating sequence {π_n}. That does not follow from (4.1) alone: weakly converging marginals can place a vanishing amount of mass at positions tending to infinity so fast that the c-integrals under π_n diverge even though (4.1) holds for the limit. Consequently Proposition 4.5 is not proved as stated. Since Theorem 6.9 invokes this stability result for the passage from compactly supported marginals to the general case, this gap compounds the moment-mismatch issue above. The proposition should either add a uniform integrability condition over ∪_n Π(µ_n,ν_n), or restrict to approximating marginals with controlled moments.
minor comments (4)
  1. [Table 2 and following paragraph] The row for (|x−x′|+|y−y′|)^2 and the sentence 'π_dia uniquely minimizes the transport cost as shown in Theorem 6.5' cite Theorem 6.5, but the relevant result is Theorem 6.2: Table 2 includes asymmetric marginals such as Exp(1), for which the symmetry assumption of Theorem 6.5 is not satisfied.
  2. [Section 7 and Appendix D] The text refers to 'Theorems 5.9' when listing closed-form results; there is no Theorem 5.9 (Definition 5.9 defines the V-transport). The intended reference is likely Theorem 5.10.
  3. [Abstract and Section 6.3] The abstract says the QOT problem is solved by the diamond transport for 'a wide class of cost functions, including the rectangular cost functions.' For the q-rectangular costs with q>1, the diamond optimality requires both marginals to be symmetric, as the paper itself emphasizes in Section 6.3; the abstract could be qualified to avoid overstating the generality.
  4. [Introduction, appendix overview] The word 'limitting' appears in the description of Appendix C; it should be 'limiting'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the diamond-transport optimality proofs are self-contained and do not reduce to fitted inputs or self-cited premises.

full rationale

Walking the derivation chain, the central claims are proved directly rather than assumed. In Theorem 6.2, the paper computes for each point (u,v) the value pi⊗pi({(u,v) in the rectangle spanned by (x,y),(x',y')}) and minimizes the resulting quadratic function in A(u,v)=pi((-inf,u]x(-inf,v]); the minimizer is then identified with the cdf of the diamond copula via the explicit formula (6.1)-(6.2). This is a closed-form optimization, not a fit and not a renaming of an input. Theorem 6.5 is proved by invoking Schoenberg's theorem and the Schur product theorem (external classical results), establishing convexity, and then using the paper's own Lemma 6.8, whose proof is carried out in Appendix G with explicit integral inequalities; no step in that proof assumes the target optimality. Theorem 6.9 uses a cost perturbation c_alpha, applies Theorem 6.5, lets alpha->0, and then invokes the stability Proposition 4.5; the stability proposition itself is proved independently using weak convergence, Sklar's theorem, and uniform integrability. The self-citations that occur (e.g., Furman et al. 2017 in Proposition C.4, Liu et al. 2025 in Example 5.5(iii), and the authors' own earlier works in examples) are auxiliary external results or contextual references; none of them is used to define the diamond transport or to establish the universal optimality claims. No fitted parameter is renamed as a prediction, and no equation equates the conclusion with an input by construction. The reviewer-flagged defect in Theorem 6.9 - that the uniform integrability bound via Cauchy-Schwarz appears to require moments of order 2q+delta while the theorem states only P_{2+delta} - is a possible correctness/stability gap in a proof step, not a circularity, because the missing moment assumption is not the conclusion of the paper and the step does not assume what it proves.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The paper uses standard results from measure theory, optimal transport, copulas, and positive definite functions. Its domain assumptions (symmetric marginals, location-scale relations, uniform marginals) are explicit scope conditions, not curve-fitting. No free parameters are fitted to data, and the new couplings are proved optimal rather than tuned.

assumptions (8)
  • standard math Sklar's theorem: every coupling has a copula representation
    Used in Proposition 4.5 and Definition 6.1 to define the diamond transport via quantile coupling.
  • standard math Schoenberg's theorem on positive definite functions and the Schur product theorem
    Invoked in Theorem 6.5 to prove convexity of the QOT cost c=phi((x-x')^2) phi((y-y')^2).
  • standard math Weak compactness of the set of couplings and the Portmanteau lemma
    Used in Propositions 4.4 and 4.5 for existence and stability of minimizers.
  • standard math Density of Monge maps in the space of couplings for atomless marginals (Santambrogio, Theorem 1.32)
    Used to prove the Monge-Kantorovich equivalence in Proposition 4.4.
  • domain assumption mu and nu are symmetric probability measures on R
    Theorems 6.5 and 6.9 require symmetric marginals for diamond transport optimality.
  • domain assumption nu is a location-scale transform of mu in Theorem 5.6
    GW-type costs are solved under this relation between marginals.
  • domain assumption mu is uniform on an interval in Theorem 5.10
    The V-transport minimizer is proven for uniform mu.
  • domain assumption Cost functions belong to C(mu,nu) and are lower semi-continuous for existence results
    Proposition 4.4 and Fact 4.3 use this integrability and regularity condition.
invented entities (2)
  • Diamond transport pi_dia, a coupling supported on the diamond D={(x,y) in [0,1]^2: |y-1/2|+|x-1/2|=1/2} in copula coordinates
    purpose: Unique minimizer for the rectangular QOT cost and for classes of type-XX costs in Theorems 6.5 and 6.9
    Defined in Definition 6.1; optimality is proven inside the paper. It has no external falsifiable prediction beyond these theorems, so it is an internally justified construction rather than an independently evidenced entity.
  • V-transport, the coupling (Q_mu(U), Q_nu(|2U-1|))
    purpose: Minimizer for separable cost f(|x-x'|)g(y,y') in Theorem 5.10
    Defined in Definition 5.9; proven optimal for a specific cost class. No external evidence outside the paper.

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Pith. "Pith review of Quadratic-form Optimal Transport." pith.science (2026). https://pith.science/paper/PK22UZC5

@misc{pith2026250104658,
  author       = {Pith},
  title        = {Pith review of: Quadratic-form Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PK22UZC5}},
  note         = {Machine review of arXiv:2501.04658}
}
abstract

We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form $\iint c\,\mathrm{d}\pi \otimes\mathrm{d}\pi$ for some coupling $\pi$ between two marginals. Interesting examples of quadratic-form transport cost and their optimization include inequality measurement, the variance of a bivariate function, covariance, Kendall's tau, the Gromov--Wasserstein distance, quadratic assignment problems, and quadratic regularization of classic optimal transport. QOT leads to substantially different mathematical structures compared to classic transport problems and many technical challenges. We illustrate the fundamental properties of QOT and provide several cases where explicit solutions are obtained. For a wide class of cost functions, including the rectangular cost functions, the QOT problem is solved by a new coupling called the diamond transport, whose copula is supported on a diamond in the unit square.

Figures

Figures reproduced from arXiv: 2501.04658 by the authors.

Figure 1
Figure 1. Illustration of the support of the comonotone transport [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The value Cdia(u, v) of the diamond copula, illustrated by distinct values in different regions. The blue shape is the support D of the diamond copula, which also indicates transitions of the cdf across different regions. 6.1 The rectangular cost function We now consider the rectangular cost function c(x, y, x′ , y′ ) = |(x − x ′ )(y − y ′ )|, which is the area of the rectangle formed by the two vertices (x, y) and … view at source ↗
Figure 3
Figure 3. Optimal coupling with quadratic-form cost function [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Plots of the QOT maximizers with cost function [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]
Figure 5
Figure 5. Figure 5: Plots of the quadratic-form optimal coupling ( [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]

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Works this paper leans on

19 extracted references · 14 canonical work pages

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    C Linear-exponential distance cost functions C.1 Basic facts All cost functions in Section 6 are symmetric in|x−x ′|and|y−y ′|. In this section, we consider a special class of type-XX cost function (a sub-class of the one treated in Theorem 5.6) that is not symmetric in|x−x ′|and|y−y ′|, which we call the class oflinear-exponential distancecost functions,...

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    The analogous question in the QOT context remains very challenging

    (i) Brenier’s theorem in classic OT states that ifµ, ν∈ P(Rd),µis absolutely continuous, and the cost is given by the squared Euclidean distance∥x−y∥ 2, then the (unique) transport plan is Monge and induced by the gradient of a convex function (Brenier [1987]; see also Santambrogio [2015, Theorem 1.17] for a more general version). The analogous question i...

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    Our next result, with a self-contained proof, implies the above conclusion onη

    showed that ifXandYare non-degenerate, thenη(X, Y) = 0 if and only ifXandYare independent, andη(X, Y) = 1 if and only ifYis a measurable function ofX. Our next result, with a self-contained proof, implies the above conclusion onη. It assumes only the first moment condition onY, much weaker than the conditions in Deb et al. [2020]. 31 Proposition C.4.Suppo...

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    λ-shaped

    Conversely, writeπ=µ⊗κ. IfE[|Y−Y ′|] = 0, thenE[|Y−Y ′| |X] = 0 almost surely, implyingµ({x: κx is degenerate}) = 1, proving that (X, Y) is Monge. The upshot of the above results is that, although Proposition B.1 implies that the independent coupling is never a minimizer for (C.2) withγ >0, we expect that the maximizersπ γ behave like the independent coup...

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    It follows from Billingsley [2013, Theorem 2.8] thatπ (n) ⊗π (n) →(π 0 −π 1)⊗(π 0 −π

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    Appendices A Quadratic programming formulation In the discrete case whereµandνare supported onNandMpoints, respectively, QOT can be formulated by a quadratic program. Denote by{x 1, . . . , xN }the support ofµand by{y 1, . . . , yM } the support ofν. Letµ i =µ({x i}) fori∈[N] ...

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