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Optimal transportation and pressure at zero temperature

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A zero-temperature pressure limit recovers optimal transport and its Kantorovich duality.

desk verdict Clean re-derivation of known entropic OT convergence, but the abstract overstates the theorem: without full-support assumptions the central claim fails. read the letter →

arxiv 2501.19369 v1 pith:46P2NH5U submitted 2025-01-31 math.DS math.FAmath.PR

Jairo K. Mengue This is my paper · ORCID
classification math.DSmath.FAmath.PR MSC 37A5028A3328D2046E2760B1060F1047H10
keywords Monge-KantorovichproblemKantorovichdualityKullback-Leiblerdivergenceentropypressurezerotemperaturelimitthermodynamicformalism
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

This paper builds a pressure function from the Kullback-Leibler divergence between a transport plan and the product of its marginals, and proves that its zero-temperature limit (β→∞) recovers the classical Monge-Kantorovich optimal transport problem together with Kantorovich duality. At each finite temperature the optimizing plan has a density of the form $e^{βA+φ_β+ψ_β}$ with respect to $μ×ν$, where $A=-c$; the paper shows such potentials exist, are essentially unique, and are Lipschitz. The main result is that the rescaled potentials $φ_β/β$ and $ψ_β/β$ converge, along subsequences, to a solution of the Kantorovich dual problem, while the plans themselves accumulate only on optimal transport plans. This matters because it connects entropic transport with thermodynamic-formalism pressure in a setting with no underlying dynamics, and it shows that the zero-temperature limit selects optimal plans of maximal relative entropy.

What carries the argument

Two devices carry the argument. First, the relative entropy $H(π)=-D_{\mathrm{KL}}(π|μ×ν)$ is a concave, upper semi-continuous functional on the compact convex set of transference plans, so the pressure supremum is attained; the entropy's variational representation converts optimality into the normalization equations, whose solution is reduced to the fixed points of the contractions $T^s_μ$ and $T^s_ν$. Second, the passage to zero temperature uses the log-sum-exp limit: Lemma 3.4 states that $(1/β)\log∫ e^{βW_β}dρ$ converges uniformly to $\sup W$ when $W_β→W$ and $ρ$ has full support, and this turns the normalization equations into the dual constraints $\sup_x(A+φ+ψ)=0$ for every $y$ and $\sup_y(A+φ+ψ)=0$ for every $x$, which are exactly the Kantorovich optimality conditions.

What would settle it

Take the finite full-support example $X=Y=\{1,2\}$, $μ=ν$ uniform, and $c(1,1)=c(2,2)=0$, $c(1,2)=c(2,1)=2$, solve the fixed-point equations for $φ_β,ψ_β$ at increasing $β$, and check whether every convergent subsequence of $(φ_β/β,ψ_β/β)$ satisfies $φ+ψ≤c$ with equality of $α(c)$ and whether every weak-* limit of $π_β$ has cost $α(c)$; a single subsequence violating the dual constraint or achieving a strictly larger cost would refute Theorem 1.1.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a Lipschitz cost $c$ and probabilities $μ,ν$ with full support, the pressure $P(βA)=\sup_{π∈Π(μ,ν)}[\int βA\,dπ + H(π)]$ with $A=-c$ and $H(π)=-D_{\mathrm{KL}}(π|μ×ν)$ has, for every $β>0$, a unique (up to an additive constant) pair $(φ_β,ψ_β)$ solving the normalization equations $\int e^{βA(x,y)+φ_β(x)+ψ_β(y)}dμ(x)=1$ for every $y$ and $\int e^{βA(x,y)+φ_β(x)+ψ_β(y)}dν(y)=1$ for every $x$. The density $dπ_β=e^{βA+φ_β+ψ_β}dμ\,dν$ is a transference plan and attains the pressure. As $β→∞$, every uniform limit $(φ,ψ)$ of $(φ_β/β,ψ_β/β)$ lies in the cone $Φ_c=\{φ(x)+ψ(y)≤c(x,y)\}$ and saturates Kantorovich duality, $α(c)=\int φ\,dμ+\int ψ\,dν$; every weak-* cluster point of $π_β$ is an optimal plan. The paper further shows that under convergence assumptions the family satisfies a large-deviation principle with rate function $c-φ-ψ$, and that the limiting plans are precisely the optimal plans with maximal relative entropy.

Load-bearing premise

The proof assumes both marginals have full support on their compact metric spaces, because the log-sum-exp limit that converts the normalization equations into the Kantorovich dual constraints is taken over the whole of $X$ and $Y$; if a marginal's support is smaller, the limiting constraint holds only on that support and an extra argument would be needed to conclude the limit pair belongs to $Φ_c$.

Editorial extensions

If this is right

  • For the distance cost on $X=Y$, any uniform limit of $φ_β/β$ solves the Kantorovich-Rubinstein dual problem with $α(c)=\int φ\,d(μ-ν)$.
  • When the limits exist, the entropic plans $π_β$ obey a large-deviation principle with rate function $c-φ-ψ$, quantifying concentration onto the optimal set.
  • All zero-temperature accumulation points of $π_β$ lie in the set of optimal plans and have the largest possible relative entropy with respect to $μ×ν$.
  • The gap $P(βA)-βm(A)$ decreases to $H_{\max}$, giving a measure of how the pressure value approaches the optimal transport value as $β→∞$.
  • The positive-temperature optimizer is an explicitly positive plan $e^{-βc+φ_β+ψ_β}μ×ν$, so the result supplies a built-in smoothing approximation to an optimal plan.

Reading between the lines

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

  • One can expect the construction to extend to costs that are merely continuous or to non-compact spaces if the log-sum-exp limit is replaced by a sup over the support, although the Lipschitz bounds used for compactness would need a substitute.
  • The maximal-entropy selection rule identifies a canonical optimal plan among many, which is the same spirit as entropic optimal transport regularization but obtained here as a zero-temperature limit rather than as a fixed small temperature.
  • If a marginal's support is a proper subset, the dual constraints are only enforced on that support; testing whether the theorem's conclusions survive with a modified normalization would clarify the essential role of the full-support assumption.
  • The rate function $c-φ-ψ$ has the form of a calibration in optimal transport, so the large-deviation result may carry over to settings such as martingale or causal transport where comparable dual potentials appear.
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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

3 major / 4 minor

Summary. The paper introduces a 'pressure' functional P(βA) for the Monge-Kantorovich problem, defined as the supremum over transference plans of ∫βA dπ + H(π), where A = −c and H is the negative Kullback-Leibler divergence relative to μ×ν. The main result, Theorem 1.1, shows that the associated Schrödinger-type normalization equations admit Lipschitz solutions (φβ, ψβ), that the Gibbs-type measures dπβ = e^{βA+φβ+ψβ}dμdν are transference plans attaining P(βA), and that as β→∞ any uniform limit of (φβ/β, ψβ/β) solves the Kantorovich dual problem while any weak-* limit of πβ solves the Monge-Kantorovich problem. The paper also derives an entropy-selection result for the limits (Proposition 1.4), a large-deviation principle for πβ (Proposition 1.3), and a Kantorovich-Rubinstein corollary when c is a distance. The proofs use contraction arguments, Arzelà-Ascoli compactness, a Laplace-principle lemma, and weak-* compactness. The paper explicitly assumes supp(μ)=X and supp(ν)=Y at the start of Section 1.

Significance. If the main theorem is correct under the stated full-support hypothesis, it gives a clean zero-temperature variational route to Kantorovich duality, connecting optimal transport with thermodynamic formalism and Sinkhorn-type normalization. The paper is largely self-contained in its central estimate, and the strategy of deriving the dual potentials and optimal plans from a single convex pressure functional is attractive. The strengths are the explicit construction of (φβ, ψβ), the relatively elementary proof of convergence, and the additional results on entropy selection and large deviations. The main concern is that the abstract states the result for arbitrary probability measures while the proof and the theorem rely essentially on full support; without it the central conclusion can fail, as the counterexample below shows. This mismatch is fixable by restating the scope, but it is load-bearing.

major comments (3)
  1. [Abstract and §1/§3.3] The abstract presents the main result for arbitrary probabilities μ∈P(X) and ν∈P(Y), but the proof is carried out under the full-support hypothesis supp(μ)=X and supp(ν)=Y, which appears only in the first paragraph of §1. This hypothesis is load-bearing: Lemma 3.4 requires supp(ρ)=U to conclude (1/β)log∫e^{βWβ}dρ → sup_U W, and in §3.3 it is applied to μ and ν to obtain equations (8) over all X and Y. If the supports are proper, the limiting equations hold only on supp(μ)×Y and X×supp(ν), so the conclusion (φ,ψ)∈Φ_c does not follow. A concrete failure is X=Y={0,1}, μ=ν=δ_0, c(x,y)=|x−y|; the normalized Schrödinger equations force φ(1)/β→1 and ψ(1)/β→1, so c(1,1)=0<φ(1)+ψ(1)=2. The abstract should either state the full-support assumption or the theorem should be reformulated for the supports of μ and ν.
  2. [§3.1 (Proposition 3.3)] The uniqueness part of Theorem 1.1 also depends on full support. In the proof that p is constant, the paper assumes p(x̃)>p0 and then uses ∫_{B(x̃,δ)} e^{A+φ1+ψ1}(e^ε−1)dμ>0; this is justified only if μ(B)>0 for every open ball, i.e., supp(μ)=X, and analogously for ν. Without full support the uniqueness claim can fail: in the δ_0 example above the normalization equations have a one-parameter family (φ(0)=t, ψ(0)=−t, φ(1)=β+t, ψ(1)=β−t), so item 1's 'unique up to a constant' statement is false. The full-support hypothesis should be recorded beside Theorem 1.1 and Proposition 3.3, not only in the opening paragraph.
  3. [§3.3] The extraction of uniform limits for φβ/β and ψβ/β needs an explicit normalization argument. The text says 'we can suppose uniformly bounded' and cites the proof of Proposition 3.1, but the bounds displayed there are for the differences φ_s − max φ_s and ψ_s − max ψ_s together with the constant l_s; they do not directly give bounds for the particular pair (φβ,ψβ) from Theorem 1.1 unless one fixes the normalization (for instance max φβ=0 and max of the auxiliary ψ before adding lβ). Please spell out this normalization and the resulting bounds, since the Arzelà-Ascoli step in item 3 depends on it. This is a local gap in presentation rather than an error in the underlying argument.
minor comments (4)
  1. [Theorem 1.1, item 1] The second Schrödinger equation is written with e^{A(x,y)+φβ(x)+ψβ(y)}, but it should be e^{βA(x,y)+φβ(x)+ψβ(y)}; without the β the statement is inconsistent with the proof and with item 2.
  2. [Proposition 3.3] In the statement of Proposition 3.3, the second equation has the quantifier '∀x∈Y'; it should be '∀x∈X'.
  3. [Proof of Proposition 3.1] After the Arzelà-Ascoli extraction, the sentence 'Particularly we get max(φβ)=max(ψβ)=0' is a typo; it should refer to the limiting φ and ψ before the shift by l, not to the β-indexed functions.
  4. [Proof of Proposition 1.4] In the final displayed conclusion, 'H(μ∞)' should be 'H(π∞)'.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: the zero-temperature limit of the entropy-regularized pressure recovers Kantorovich duality by direct limit arguments; the imports from prior work are independent lemmas, not the target conclusion.

full rationale

The paper's derivation chain is direct. It defines a pressure P(βA) using relative entropy, constructs Schrödinger-type potentials φ_β, ψ_β via a contraction argument (Section 3.1), proves finite-temperature duality P(βA) = -∫φ_β dμ - ∫ψ_β dν (Section 3.2), and then passes β → ∞ using the Laplace principle (Lemma 3.4) to derive the limiting equations sup_x A(x,y)+φ(x)+ψ(y)=0 and sup_y A(x,y)+φ(x)+ψ(y)=0, which yield (φ,ψ) ∈ Φ_c and the Kantorovich duality equality (Section 3.3). The two imported results, the entropy variational formula from [8] (equation (1)) and Lemma 3.4 from [11], are published theorems in their own right; neither states or assumes the target Kantorovich duality, and the paper does not rename a fitted parameter as a prediction. The central claim is obtained by taking limits of constructed objects, not by assuming the conclusion. There are correctness caveats, not circularity: the proof explicitly assumes supp(μ)=X and supp(ν)=Y (Section 1), which Lemma 3.4 and the uniqueness proof (Proposition 3.3) genuinely require; the uniform-boundedness normalization in Section 3.3 is sketched rather than fully justified; and the second equation in Theorem 1.1 item 1 appears to contain a typo (exponent A instead of βA, contradicted by item 2 and Section 3.3). These are proof-rigor issues that would matter for the theorem's advertised scope, but they do not make the derivation circular.

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

No free parameters are fitted. The paper introduces no new entities. It relies on standard functional analysis and on two imported results from the author's prior work (entropy representation and Laplace principle), plus the full-support and Lipschitz assumptions on the input data. The proof of the central theorem is otherwise self-contained.

assumptions (6)
  • standard math Compact metric spaces X,Y with Borel sigma-algebra; weak* compactness of probability measures on compact spaces; Arzelà-Ascoli theorem.
    Background used throughout, for example to extract convergent subsequences of (π_β) and of normalized potentials in Sections 2 and 3.
  • domain assumption supp(µ)=X and supp(ν)=Y.
    Assumed in the first paragraph of Section 1; required for Lemma 3.4 to yield sup over the whole space in the zero-temperature limit. Without full support, the conclusion that (φ,ψ) belongs to Φ_c would not follow from the proof.
  • domain assumption A = -c is Lipschitz continuous on X×Y.
    Ensures the operators T^s in Proposition 3.1 are well-defined and estimated, that φβ and ψβ are Lipschitz, and that the family (φβ/β, ψβ/β) is equicontinuous.
  • domain assumption Entropy representation H(π) = -sup{∫ u dπ | ∫ e^u dµ = 1 ∀y, u Lipschitz} (equation (1), from [8]).
    Quoted from the author's prior work with Lopes; used in the proof of item 2 of Theorem 1.1 to show P(βA) ≤ -∫φβ dµ - ∫ψβ dν. It is a published theorem, not proved in this paper.
  • standard math Lemma 3.4 (Laplace principle): for Wβ → W uniformly, (1/β) log ∫ e^{β Wβ} dρ → sup W, provided supp(ρ) is the whole space.
    Imported from [11] by the same author; used to pass from the Schrödinger normalization equations to the sup equations (8) in Section 3.3.
  • domain assumption For Proposition 1.3, existence of the uniform limits φ = lim φβ/β and ψ = lim ψβ/β and of the weak* limit π = lim πβ is assumed.
    The LDP is stated conditionally ('supposing there exist'); the paper only proves subsequential limits, not full convergence, so this is an added premise.

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Pith. "Pith review of Optimal transportation and pressure at zero temperature." pith.science (2026). https://pith.science/paper/46P2NH5U

@misc{pith2026250119369,
  author       = {Pith},
  title        = {Pith review of: Optimal transportation and pressure at zero temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46P2NH5U}},
  note         = {Machine review of arXiv:2501.19369}
}
abstract

Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $\mu \in\mathcal{P}(X),\,\nu\in\mathcal{P}(Y)$, we propose to study the Monge-Kantorovich problem and its duality from a zero temperature limit of a convex pressure function. We consider the entropy defined by $H(\pi) = -D_{KL}(\pi|\mu\times \nu)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(\beta A) = \sup_{\pi \in \Pi(\mu,\nu)} \left[ \smallint \beta A\,d\pi + H(\pi)\right],\]where $\beta>0$ and $A=-c$. We will show that it admits a dual formulation and when $\beta \to+\infty$ we recover the solution for the usual Monge-Kantorovich problem and its Kantorovich duality. Such approach is similar to one which is well known in Thermodynamic Formalism and Ergodic Optimization, where $\beta$ is interpreted as the inverse of the temperature ($\beta = \frac{1}{T}$) and $\beta\to+\infty$ is interpreted as a zero temperature limit.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Entropic optimal transport need not select a zero-temperature limit

    math.OC 2026-07 accept novelty 8.0 of 10

    Entropic optimal-transport minimizers need not converge as ε→0 even for compact, atomless, bounded-Lipschitz costs; the cluster set can be an interval of optimal plans.

Reference graph

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