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Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read SOS certificates now prove where optimal transport maps stay regular

desk verdict The forward SOS certification of MTW/NNCC is new and usable; the inverse problem is invalid as stated because Theorem 7 relies on a false principal-minor criterion and Theorem 8 returns the wrong kind of set. read the letter →

arxiv 2412.13372 v1 pith:ARMMCWZ4 submitted 2024-12-17 math.OC cs.AIcs.LGmath.DG

Sachin Shivakumar , Georgiy A. Bondar , Gabriel Khan , Abhishek Halder This is my paper · ORCID
classification math.OCcs.AIcs.LGmath.DG MSC 49Q2290C2214P10
keywords optimaltransportMa-Trudinger-WangtensorMTWconditionMongemapregularitysum-of-squaresprogrammingsemidefinitenon-negativecostcurvaturesemialgebraicgeometry
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

The paper sets out to make the Ma–Trudinger–Wang (MTW) condition algorithmically checkable: for a given ground cost, it wants to certify or falsify non-negativity of the fourth-order MTW tensor, and to compute the region in which that non-negativity holds. The proposed route is sum-of-squares (SOS) programming. Whenever the MTW tensor matrix is rational over a semialgebraic domain, the certification problem becomes an SOS feasibility problem, hence a semidefinite program, via Theorems 5 and 6. The inverse problem is solved by parameterizing the unknown region as the zero sublevel set of a polynomial and minimizing its volume under SOS constraints, yielding inner approximations claimed by Theorems 7 and 8. Numerical examples recover a known analytical threshold for a perturbed Euclidean cost, certify a log-partition cost in dimensions three through six, and localize regularity for a positive-curvature squared-distance cost.

What carries the argument

The central object is the MTW tensor matrix $F(x,y)\in\mathbb{R}^{n^2\times n^2}$ built from partial derivatives of the ground cost $c$ and from the inverse mixed Hessian $H=((\nabla_x\otimes\nabla_y)c)^{-1}$; the MTW quadratic form is $(\xi\otimes\eta)^\top F(\xi\otimes\eta)$. The carrying mechanism is the Positivstellensatz in its SOS form: a polynomial that is non-negative on an Archimedean semialgebraic set can be written as a sum of squares plus multipliers of the set-defining inequalities. The forward theorems substitute that representation into the MTW inequality, turning it into a semidefinite program; the inverse theorems use the same representation with the region-defining polynomial $V$ as the decision variable, so the zero sublevel sets of $V_+$ and $V_-$ give the inner approximation of the regular region.

What would settle it

Apply the $V_-$ argument to $F_N=\begin{pmatrix}-1&2\\2&-1\end{pmatrix}$ with $F_D=-1$: every principal minor of $F_N$ is non-positive, so the proof's premise is satisfied, yet $F_N$ has a positive eigenvalue and $-F_N$ is not positive semidefinite; a feasible $V_-$ SOS certificate would therefore certify NNCC where it does not hold.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that optimal-transport regularity verification reduces to linear algebra: the MTW tensor is written as $S(x,y)(\xi,\eta)=(\xi\otimes\eta)^\top F(x,y)(\xi\otimes\eta)$, and when the entries of $F$ are rational functions, the condition $S\ge 0$ on a semialgebraic set is equivalent to an SOS feasibility problem. The same reduction handles the strong MTW($\kappa$) condition, with the orthogonality constraint $\eta(\xi)=0$ absorbed by a polynomial multiplier in the SOS identity. For the inverse problem, the paper claims that an inner approximation of the regular region can be computed by minimizing the volume of a polynomial sublevel set subject to SOS constraints, with two certificates $V_+$ and $V_-$ covering the cases where the denominator of $F$ is positive or negative. The paper therefore claims the first provably correct computational framework for certifying, falsifying, and localizing MTW and non-negative cost curvature (NNCC) regularity for a broad class of ground costs, including costs that are not themselves rational as long as $F$ is rational.

Load-bearing premise

The proof of the inverse theorem's negative-denominator branch assumes that a symmetric matrix whose principal minors are all non-positive must be negative semidefinite, an implication that is false in dimension two and higher.

Editorial extensions

If this is right

  • For rational ground costs, MTW(0) and MTW($\kappa$) verification becomes a semidefinite program; feasibility is a certificate of regularity on the entire semialgebraic domain.
  • Infeasibility of the forward SOS program is a computational falsification of MTW or NNCC on that domain, giving a concrete breakdown point for continuity of the Monge map.
  • The inverse program outputs an explicit polynomial $V$ whose zero sublevel set is a provable inner approximation of the regular region, so the user knows exactly where the map is guaranteed to be continuous.
  • The method reproduces analytically known results, such as the $\varepsilon_{\max}=2/3$ threshold for the one-dimensional perturbed Euclidean cost, and extends to log-partition costs whose MTW tensor is rational even though the cost itself is not.
  • Because the certificates are SOS identities, the outputs are machine-checkable and can be handed to existing semidefinite-programming solvers.

Reading between the lines

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

  • Inference: the forward certificate could be used as a design constraint on a parameterized family of costs, letting one search over costs that are guaranteed MTW-regular rather than merely checking a fixed cost.
  • Inference: the paper notes that MTW and NNCC imply connected $c$-subdifferentials and faster $c$-conjugation; a testable extension is to use the computed regions as domain decompositions inside numerical OT solvers, where continuity of the map is known a priori.
  • Inference: for non-rational costs such as $-\log\|x-y\|$, rational approximation of the cost would make the same pipeline applicable to reflector-antenna and lens-design problems; the paper lists these as motivating examples but does not run them.
  • Inference: bisection in $\kappa$ over the forward MTW($\kappa$) certificate would yield the largest strong-MTW constant, turning the binary certificate into a quantitative regularity margin.
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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

4 major / 6 minor

Summary. The paper proposes a sum-of-squares (SOS) programming framework for two tasks in optimal transport regularity: (i) certifying non-negativity of the Ma-Trudinger-Wang (MTW) tensor / non-negative cost curvature (NNCC) for a given rational ground cost over a semialgebraic domain, and (ii) computing semialgebraic inner approximations of the region where these conditions hold. The forward method is stated in Theorems 5 and 6, the inverse NNCC method in Theorem 7, and the inverse MTW method in Theorem 8. Numerical examples treat a perturbed Euclidean cost, a log-partition cost for the isotropic multivariate normal, and a squared-distance cost for a positively curved surface, using SOSTOOLS/YALMIP.

Significance. If the proposed theorems were correct, the paper would contribute a broadly applicable computational tool to a problem—verifying MTW/NNCC conditions—that has previously been handled only by hand-crafted analytic arguments for specific costs. The forward SOS certificates are attractive because they reduce a hard nonnegativity question to an SDP, and the inverse certificates would give a constructive way to localize regularity regions. The numerical experiments are concrete, and the paper honestly reports residuals and timings. However, the advertised 'provably correct' claim is compromised by several load-bearing gaps: the forward theorems omit a sign condition on the denominator, the inverse NNCC theorem relies on a false matrix-semidefiniteness criterion, and the inverse MTW theorem returns a subset of the extended (x,y,xi,eta) space rather than a subset of X×Y. These issues need substantial repair before the central claims are supported.

major comments (4)
  1. [Section 3.1, Theorem 5 (and Theorem 6)] The proof of Theorem 5 infers from the certificate (13) that F_N(x,y)+F_N^T(x,y) is positive semidefinite only under the additional condition F_D(x,y) >= 0. When F_D < 0 on a subset of X×Y, the certificate imposes no useful lower bound on F_N; indeed, for the scalar rational matrix F = -1 represented as F_N = 1, F_D = -1, the choice s0 = 0 satisfies (13) with no constraints, yet NNCC fails. Thus Theorem 5 is not a sound sufficient condition as stated. Theorem 6 has the same sign deficiency: for F_D < 0, inequality (14) implies an upper bound on the quadratic form of F_N after division by a negative denominator, which is the opposite of the MTW lower bound. The statements should either explicitly assume F_D > 0 on X×Y or split into the two sign cases handled later in Theorem 7.
  2. [Section 3.2, Theorem 7, V− branch] In the V− branch, the SOS constraints together with V(x,y) <= 0 imply that all principal minors f_j of F_N satisfy f_j <= 0. The proof then concludes that F_N is negative semidefinite, so that F_N/F_D is positive semidefinite when F_D < 0. This implication is false for n >= 2: the matrix A = [[-1, 2], [2, -1]] has all principal minors non-positive but is indefinite (eigenvalues 1 and -3). Consequently the claimed inner approximation {(x,y) ∈ Λ | V−(x,y) <= 0} is not certified to be a region where NNCC holds. The correct condition for negative semidefiniteness requires all principal minors of -F_N to be nonnegative, i.e., the principal minors of F_N of size k must have sign (-1)^k; non-positive principal minors alone are insufficient.
  3. [Section 3.2, Theorem 8] The inverse problem (16) asks for a subset U×V of X×Y on which the MTW(κ) condition holds, but Theorem 8 returns a subset of the extended variable space: {(x,y,ξ,η) ∈ Λ | V±(x,y,ξ,η) <= 0, η(ξ)=0}. This is not a subset of X×Y, and even if a base point (x,y) belongs to the projection, the theorem only certifies the inequality for those (ξ,η) that lie in the sublevel set, not for all orthogonal pairs (ξ,η) as required by MTW(κ). Therefore the regions plotted in Examples 3 and 4 (Figures 1 and 2) are not established as regions of MTW regularity in the sense of the advertised inverse problem. The theorem needs to be reformulated so that V is independent of (ξ,η) (or otherwise quantified over all admissible pairs) and the returned object is a genuine base-space region.
  4. [Introduction and Section 3.2] The paper repeatedly states that the framework can 'certify or falsify' non-negativity of the MTW tensor (Abstract and Contributions), and Section 3.2 says infeasibility of the forward problem 'falsifies' the condition. However, the numerical pipeline solves SOS tightenings, which are only sufficient conditions; infeasibility of the SOS program at a given relaxation degree does not imply that the original polynomial inequalities are violated. The bisection estimates in Section 4.1 therefore do not rigorously establish the reported thresholds εmax as exact bounds. The language should be softened to 'certify' only, or the falsification claim should be justified with a convergent hierarchy or an explicit separating point.
minor comments (6)
  1. [Section 3.2, Eq. (15)-(19)] The notation U×V is used for an arbitrary sublevel set {(x,y) ∈ X×Y | V(x,y) <= 0}, which is not necessarily a Cartesian product of a subset U of X and a subset V of Y. This is misleading; consider using a different symbol such as W or R for the candidate region.
  2. [Theorems 7 and 8 statements] The use of '±' in the constraints is terse: the plus and minus signs correspond to two separate SOS systems for V+ and V−, respectively. The statement would be clearer if the two systems were written out explicitly.
  3. [Table 2 and Table 3] The definition of the residual as 'the largest coefficient in the polynomial S − s^T s' is unclear: S and s are not precisely defined in the table caption, and it is not evident that the quantity is nonnegative or that it measures feasibility error. Please spell out the residual computation.
  4. [Appendix D] The reported SOS decomposition for the n=3 log-partition example presents s as a matrix of coefficients multiplied by a monomial vector, but the surrounding text writes s(x,ξ,η)^T s(x,ξ,η). Please clarify whether s is a vector or a matrix and whether the displayed object is the Gram-matrix factor.
  5. [Section 4.1, Example 1] The sentence 'For n = 1, there are no pairs ξ, η such that η(ξ) = 0 with η and ξ both non-zero' is correct, but it implies that MTW(0) is vacuous in 1D; the example appears to be verifying NNCC instead. The text should state this explicitly to avoid confusion.
  6. [Throughout] There are numerous typographical artifacts in the rendering: for example, 'Y ALMIP' appears with an internal space, 'Positivstellansatz' is misspelled, and 'X Y' is sometimes printed without a product symbol. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the SOS certifiability results follow from Putinar's Positivstellensatz and the examples are validated against independent analytic results; self-citations are motivational only.

full rationale

The paper's central derivations are self-contained in the relevant sense. Theorem 5 translates the NNCC condition F = FN/FD ⪰ 0 into the existence of matrix SOS multipliers satisfying (13); the proof only uses the elementary fact that a sum-of-squares multiplier is nonnegative on the described semialgebraic set, and the conclusion F ⪰ 0 follows directly from the certificate. Theorem 6 does the same for the constrained MTW(kappa) inequality using the eta^T xi multiplier. Neither theorem imports the MTW/NNCC conclusion as an assumption, and no fitted parameter is renamed as a prediction. Example 1 validates the SOS bisection against the independently known analytic threshold eps_max = 2/3. Example 2 cites Khan and Zhang (2020, 2022), including work by a coauthor, for the fact that A >= 0 implies regularity of the log-partition cost, but the SOS certificate itself proves nonnegativity of poly(x,xi,eta) without invoking that theorem; the citation is interpretive, not load-bearing. The inverse theorems (7 and 8) are not circular either: they solve for a polynomial V whose sublevel set is constrained to lie inside the NNCC/MTW region, and the conclusion restates the enforced feasibility constraints. There is, however, a serious correctness gap in the V- branch of Theorem 7, Section 3.2: the proof treats 'all principal minors of FN are <= 0' as implying FN is negative semidefinite, which is false for n >= 2 (e.g., [[-1,2],[2,-1]]). This is a mathematical error, not a circularity, and it does not make the derivation equivalent to its inputs. Likewise Theorem 8 returns a sublevel set in the extended variables (x,y,xi,eta) rather than a product set U x V subset of X x Y, but that is a statement mismatch, not a circular reduction. Overall, the advertised forward certificates have independent content; the score reflects only the presence of coauthor citations that are not load-bearing.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The forward theorems rely only on standard SOS and Positivstellensatz facts. The inverse problem adds choices of degrees and Lambda, a volume surrogate heuristic, and contains a false assumption about principal minors. No new physical entities are introduced.

free parameters (2)
  • SOS certificate degrees
    The theorems require choosing polynomial degrees for the SOS multipliers; no automated degree selection is given. In Examples 3 and 4, V is fixed to degree 14 by hand. The existence and quality of certificates depend on these choices.
  • Integration domain Lambda = Examples: [-1,1]^2 and [-1,1]x[0,2]
    The compact set Lambda is chosen a priori and affects the computed inner approximation; the paper gives no principled selection rule.
assumptions (6)
  • standard math Putinar's Positivstellensatz
    Used in Section 2.2 to replace nonnegativity over Archimedean sets by an SOS representation, as in equation (8).
  • standard math Archimedean property can be achieved by adding a ball constraint
    Invoked in Appendix A.3 to justify that compact semialgebraic sets can be made Archimedean for SOS verification.
  • standard math Sylvester's criterion for positive semidefinite matrices
    Used in the proof of Theorem 7 to conclude that nonnegativity of all principal minors of FN implies FN is positive semidefinite.
  • ad hoc to paper All principal minors non-positive implies negative semidefinite
    Invoked in the V- part of Theorem 7's proof; this implication is false for n at least 2.
  • ad hoc to paper Minimizing the integral of V is a valid surrogate for maximizing the volume of the inner approximation
    Used in Section 3.2 to convert the volume maximization into an SOS program, citing a heuristic from Jones (2024); no proof of equivalence or approximation guarantee is given.
  • domain assumption A sublevel set in (x,y,xi,eta) space can be interpreted as a region U times V in X times Y
    The inverse problem statement promises a region in the base space, but Theorems 7 and 8 certify subsets of the extended path space.

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Cite this review

Pith. "Pith review of Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps." pith.science (2026). https://pith.science/paper/ARMMCWZ4

@misc{pith2026241213372,
  author       = {Pith},
  title        = {Pith review of: Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARMMCWZ4}},
  note         = {Machine review of arXiv:2412.13372}
}
read the original abstract

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.

Figures

Figures reproduced from arXiv: 2412.13372 by the authors.

Figure 1
Figure 1. Inner approximation of the region where MTW tensor is ≥ 0 for Example 3. Recall from Sec. 4.1 that the cost c(x, y) = ∥x − y∥ 2 − ε∥x − y∥ 4 fails the MTW(0) condition for large ε > 0. To illustrate the solution for the MTW(0) in￾verse problem (Sec. 3.2), we fix ε = 1, Λ = [−1, 1]2 , X = {[0, 0]}, and X × Y as in Example 1. As be￾fore, we parameterize (ξ, η) as ξ = [a, 1]⊤, η = [−1, a] ⊤. We solve the SOS formulatio… view at source ↗
Figure 2
Figure 2. Inner approximation of the region where MTW tensor is ≥ 0 for Example 4. This metric has two further properties which make analysis of its MTW tensor more tractable. First, it admits a symmetry in the x1 coordinate, which allows us to set x1 = 0. Second, the scalings (x1, x2) 7→ (ax1, ax2) are homotheties of the metric. Therefore, we can scale the met￾ric so that x2 = 1, and so to determine regions for which the MTW… view at source ↗
Figure 3
Figure 3. The Motzkin polynomial. That polyMotzkin does not admit a SOS rep￾resentation, which can be proven by contradiction. Specifically, suppose that polyMotzkin(x1, x2) = Xm i=1 (polyi (x1, x2))2 where poly1 , poly2 , . . . , polym ∈ R[x1, x2] for some m ∈ N. Comparing the terms of same to￾tal degree from both sides, we find the degree of 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparing contour plots of various ground costs [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.