Entropic optimal transport need not select a zero-temperature limit

Maja Gwozdz

REVIEW 3 major objections 4 minor 23 references

Entropic optimal transport need not select a zero-temperature limit

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Entropic optimal transport need not select a zero-temperature limit: the author constructs a compact atomless example with bounded Lipschitz cost in which the entropic minimisers oscillate and never converge as ε↓0.

desk verdict The connectedness theorem is solid and worth having, but the central nonconvergence proof has a false asymptotic tail estimate and the oscillation claim is not established as written. read the letter →

arxiv 2607.16881 v1 pith:S6URNN3H submitted 2026-07-18 math.OC math.PR

Maja Gwozdz This is my paper · ORCID

classification math.OCmath.PR MSC 49Q2260B1060E05

keywords entropicoptimaltransportSchrödingerbridgezero-temperaturelimitnonconvergenceselectionprincipleclustersetdiagonalscalingboundedLipschitzcost

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 tries to establish that entropic optimal transport can have no zero-temperature limit, even in a setting as regular as one might hope: a compact metric space, an atomless probability measure (no single point carries positive mass), and a bounded Lipschitz cost. The author constructs an explicit dyadic Cantor-group example in which the entropic minimisers P_ε do not converge weakly as ε↓0. Instead, all subsequential limits form a non-degenerate continuum of optimal plans, each a mixture of the two zero-cost graph couplings. A companion theorem shows that in all Polish settings with continuous integrable costs, the zero-temperature cluster set is always a nonempty weakly compact connected subset of the optimal face, so the failure is not wild: it happens exactly through a connected continuum. A reader should care because the conventional picture—that entropic regularisation selects a distinguished optimal plan in the zero-temperature limit—is not generally valid, and the obstruction appears at the level of basic regularity assumptions.

What carries the argument

The load-bearing construction is the dyadic Cantor-group model: X is the compact group {0,1}^N with the ultrametric, μ is the atomless product of fair-coin measures, and the cost is C(x,y)=γ(x)W(y−x), whose zero set is Z={y−x∈{0,t_0}}. The proof reduces the entropic minimisation to a 2×2 diagonal-scaling problem for a block kernel; the scalar w_ε, which measures the weight the minimiser puts on the diagonal graph, oscillates along ε_n=e^{-2n} between two limits determined by alternating scale factors. The general connectedness theorem—the cluster set is a nonempty weakly compact connected subset of the optimal face—then upgrades the two oscillation limits to an entire interval of cluster poi

What would settle it

Compute the scalar w_ε = P_ε({(x,y):(y−x)_1=0}) along ε_n = e^{-2n}; the theorem predicts it alternates between the two closed-form limits w±. If a numerical or exact evaluation shows w_ε has only one limit, or if some other plan outside the segment {P_w} appears as a cluster point, the central nonconvergence claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is a counterexample: there exist a compact metric space X, an atomless probability measure μ, and a bounded Lipschitz cost C:X×X→[0,∞) such that the entropic minimisers P_ε∈Π(μ,μ) do not converge weakly as ε↓0. In the construction, the zero-cost set is the union of the diagonal and one translate of the diagonal, and every unregularised optimal plan is singular with respect to μ⊗μ, so the relative entropy on the optimal face is identically +∞. The cluster set is exactly {P_w = wP_0+(1−w)P_{t_0} : w∈W}, where W⊂[0,1] is a non-degenerate compact interval; the paper computes two explicit limits w− < w+ from alternating temperature scales. It also proves a general compac

Load-bearing premise

The proof that every weak cluster point is an optimal transport plan relies on a previously established cluster-point theorem that the paper invokes as a black box; if that theorem carries an additional integrability or regularity hypothesis not satisfied by the dyadic Cantor cost, the identification of the cluster set as the interval {P_w} would need to be reworked.

Editorial extensions

If this is right

Zero-temperature convergence fails under compactness, atomlessness, and bounded-Lipschitz costs; any positive selection theorem for entropic optimal transport needs additional structure beyond these assumptions.
In general, the zero-temperature cluster set is always a nonempty weakly compact connected subset of the optimal face, so nonconvergence can only occur through a connected continuum of optimal plans, never through isolated oscillations.
In the example, every optimal plan is singular with respect to the product measure, so the entropy functional is +∞ on the entire optimal face; the entropy term cannot serve as a tie-breaker.
The variational criteria give first-order conditions: a plan is a cluster point iff a certain local gap divided by ε has liminf 0, and full convergence holds iff every exterior gap has positive liminf as ε↓0.
The explicit weights w± provide a concrete pair of distinct subsequential limits, demonstrating the oscillation quantitatively.

Reading between the lines

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

The oscillation mechanism suggests a two-scale test: if a cost's zero set is a 'fork' with two graphs and the positive values decay at exponentially separated scales with parity-dependent prefactors, similar nonconvergence may be constructible; one could try to build a one-dimensional analogue on [0,1].
The connectedness theorem implies that a cheap way to prove convergence in a specific model is to show the optimal face is a singleton at any cluster point; then compactness and connectedness force the whole cluster set to be one point, so convergence follows without further estimates.
The variational criteria are computable in principle: for a proposed optimal plan, one can numerically estimate the normalized local gap near it; a positive lower bound for all exterior radii would certify convergence, while a zero value certifies cluster membership.
Because the counterexample lives on a Cantor group, one might guess that Euclidean regularity (e.g. C^1 or Hölder costs on manifolds) restores selection; testing whether the alternation survives under such regularity is a natural next question left open by the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the general compactness/connectedness theorem is correct and valuable, but the counterexample proof as written has a real bug in the asymptotic tail estimate. The paper should not be accepted in this form; Lemma 6.1/Claim 6.2 needs to be repaired before the main theorem can be believed.

What is new and good: the paper identifies a natural open question—whether compact, atomless, bounded-Lipschitz costs force zero-temperature convergence—and gives a plausible counterexample mechanism via a dyadic Cantor-group cost. The connectedness theorem (Clust(Pε) is a nonempty compact connected subset of the optimal face under continuity, L1-bounded-below costs) is clean, and the proof via nested closures of continuous images is sound. The reduction to a 2×2 Sinkhorn scaling, the exact series representations, and the classification of diagonally H-invariant zero-cost couplings are all carefully done. I also checked the reader's concern about relying on the BGN cluster-point theorem: the finite-value hypothesis is checked with Λ, so that is not a serious gap.

The soft spot is in Section 6. Lemma 6.1/Claim 6.2 asserts that along ε_n = α_n, the m>n tail is o(2^{-n}). The proof bounds α_m/ε_n = e^{-2(m-n)} ≤ e^{-2n}, which is false: for m=n+1 the ratio is e^{-2}, not e^{-2n}. The tail is of order 2^{-n} with a nontrivial constant, roughly Σ_{k≥1}2^{-k}e^{-b_{n+k}e^{-2k}} ≈ 0.932 for b≡1. So I0(ε_n) is not 2^{-n}(1+e^{-1}) but 2^{-n}(e^{-1}+0.932)+o(2^{-n}). The same issue affects I1, J0, J1. Consequently the closed forms for w± in Corollary 6.3 are not justified, and the monotonicity argument in Proposition 6.4 cannot be verified. It is possible the construction still oscillates—the alternation of L_n may still yield distinct limits once the tail is computed correctly—but that remains to be shown. The reader's soundness score of 8 is too generous on this point.

The audience is anyone working on zero-temperature limits of entropic optimal transport or Sinkhorn selection. The connectedness theorem alone justifies a serious referee, but the counterexample is the headline, and that part is currently unproved. My recommendation: send it to peer review, but be explicit that the asymptotic computation must be fixed and the oscillation checked with correct tail constants. As it stands, I would not cite the main theorem.

Referee Report

3 major / 4 minor

Summary. The paper studies the zero-temperature limit of entropic optimal-transport minimizers. Its main general result (Theorem 1.1/3.1) states that, for continuous costs that are bounded below and integrable with respect to the product reference, the cluster set of the entropic minimizers as ε↓0 is a nonempty weakly compact connected subset of the optimal face. The paper then constructs a dyadic Cantor-group example (Theorem 1.2/6.6) with compact space, atomless marginals, and a bounded Lipschitz cost, for which the entropic minimizers do not converge weakly. The cluster set is identified as a non-degenerate compact interval of mixtures of two zero-cost graph couplings, and two explicit distinct weights w−<w+ are claimed. The proof combines the Bernton–Ghosal–Nutz cluster-point theorem, a continuity argument for the entropic trajectory, a reduction of the Schrödinger projection to a 2×2 Sinkhorn scaling, and an asymptotic analysis along ε_n=α_n=e^{-2n}.

Significance. If the construction is correct, the paper disproves a natural selection principle: compactness, atomlessness, and Lipschitz regularity of the cost do not force convergence of entropic minimizers at zero temperature. The general connectedness theorem is a valuable contribution in itself, and the paper also provides local and exterior variational criteria for cluster membership and full convergence. The Cantor-group example is elegant and the reduction to a scalar branch weight is conceptually appealing. The manuscript presents detailed proofs of the Lipschitz estimate, the 2×2 scaling identities, and the invariance classification. However, the central counterexample relies on an asymptotic lemma (Lemma 6.1) whose proof contains a genuine error; the claimed limits are not established as written. The construction may be salvageable, but the present proof of nonconvergence is incomplete.

major comments (3)

[§6.1, Claim 6.2 and Lemma 6.1] The proof of Claim 6.2 contains a false inequality. It states that for m>n, α_m/ε_n = exp(−(2m−2n)) and therefore 0 ≤ α_m/ε_n ≤ e^{-2n}. But for m=n+1 the ratio is e^{-2}, which is larger than e^{-2n} for n≥2. Consequently the bound |e^{-b_m α_m/ε_n}−1| ≤ b_* e^{-2n} is invalid. The m>n tail contributes a term of order 2^{-n} with constant Σ_{k≥1} 2^{-k} e^{-b_{n+k} e^{-2k}}, not o(2^{-n}). Thus the leading asymptotics in Lemma 6.1 (e.g., I0(ε_n)=2^{-n}(1+e^{-1})+o(2^{-n})) are not justified. The same flaw propagates to I1,J0,J1 and to the formula for w_{ε_n}.
[Corollary 6.3 and Proposition 6.4] The closed forms for w± and the proof that w+>w− rest on the incorrect asymptotic of Lemma 6.1. Since the tail constants depend on the full sequence L_{n+k} for k≥1, not only on L_n, the limits along even and odd n may have different constants, but the manuscript does not compute them. The strict monotonicity argument in Proposition 6.4 is not applicable to the actual leading-order terms. Without distinct limits w+ and w−, the non-degeneracy of the interval W and therefore the nonconvergence statement in Theorem 6.6 are not established.
[Theorem 1.2 / Theorem 6.6] Because the only mechanism producing two distinct cluster points is the asymptotics of w_{ε_n}, the main theorem is not proved as written. The construction may still work, but a correct asymptotic computation is required. This is a load-bearing issue, not a presentation matter.

minor comments (4)

[§6.1, around (6.4)] The statement that o(2^{-n}) terms are uniform for L_n∈[1−a,1+a] is misleading: even if the error in each tail term were uniform, the tail constant itself is a function of the sequence (L_{n+k}). The uniformity claim does not repair the missing tail estimate.
[Proposition 6.5 proof] Typo: 'convering' should be 'converging' in the sentence '...has another subsequence convering to Pw'.
[§5.3, after (5.6)] The matrix Kε is displayed as (I0 I1; J1 J0) with rows x1∈{0,1} and columns y1∈{0,1}; the ordering of rows/columns is not explicitly stated but is clear from the subsequent formulas. Consider adding one sentence to avoid confusion.
[§5.3, Lemma 5.5] The final sentence of Lemma 5.5 ('The formula follows by checking the marginal constraints and then applying the Csiszár projection criterion.') appears to belong to the next proposition rather than to the invariance lemma. It would be clearer to move it to the proof of Proposition 5.6.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the counterexample's quantities are derived, not fitted, and no load-bearing self-citation occurs.

full rationale

The derivation chain is self-contained. The scalar branch weight w_epsilon is defined explicitly in (5.4) as A_epsilon/(A_epsilon+B_epsilon) from the four partition functions I0,I1,J0,J1, and the alternating limits w+ and w- are computed in Corollary 6.3 from the asymptotics of those functions; they are not fitted to a target. The cluster set Clust(P_epsilon) = {P_w : w in W} is obtained in Proposition 6.5 by combining the explicit formula for the entropic minimiser (5.7), the optimality of cluster points (imported from Bernton–Ghosal–Nutz [2] as an external theorem, with the finite-value hypothesis verified using the product coupling Lambda), and the classification Lemma C.1 of H-invariant zero-cost couplings. The interval structure of W is imported from the connectedness theorem applied to the cluster set, not assumed. There are no self-citations by the present author and no parameter is fitted to the quantity it is said to predict. Any challenge to Lemma 6.1's tail estimate would be a mathematical-error objection, not a circularity objection; the main theorem's derivation does not reduce to its inputs by definition.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on standard OT/entropy facts plus one external cluster-point theorem; no data fitting or fitted constants are involved. The free parameters a, β, α_n parameterize the explicit counterexample family.

free parameters (3)

a (alternation amplitude) = a∈(0,1), arbitrary
Defines L_n=1+a for even n and 1−a for odd n; controls the separation w+−w−. The theorem holds for every a∈(0,1); not fitted.
β (cost asymmetry) = β∈(0,∞)\{1}, arbitrary
Sets γ=1 on G0 and γ=β on G1; ensures the Gibbs reference R_ε is not already a coupling. Any β≠1 works; not fitted.
scale sequence α_n = α_n=e^{-2n}
Chosen in the model definition (4.1); the double-exponential decay lets the n-th cylinder dominate at ε=α_n. It is a construction choice, not fitted to a target.

assumptions (5)

domain assumption BGN cluster-point theorem [2, Props 2.2 and 3.2]: weak cluster points of entropic minimizers are optimal under the finite-value condition
Imported in Prop 3.3; the paper verifies the finite-value condition with Λ.
standard math Relative entropy is weakly lower semicontinuous [10]
Used in Prop 2.1 and Prop 3.5.
standard math Pinsker's inequality
Used in Lemma A.4.
standard math Haar measure on compact group G is translation-invariant and atomless
Used throughout Sections 4–5.
standard math Strict convexity of KL and uniqueness of the entropic minimizer
Used in Prop 2.1.

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Cite this review

Pith. "Pith review of Entropic optimal transport need not select a zero-temperature limit." pith.science (2026). https://pith.science/paper/S6URNN3H

@misc{pith2026260716881,
  author       = {Pith},
  title        = {Pith review of: Entropic optimal transport need not select a zero-temperature limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6URNN3H}},
  note         = {Machine review of arXiv:2607.16881}
}

abstract

We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $\mu\otimes\mu$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(\mu\otimes\nu)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapsto\pi_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Entropic selection principle for Monge’s optimal transport, 2025

Shrey Aryan and Promit Ghosal. Entropic selection principle for Monge’s optimal transport, 2025. Preprint, arXiv:2502.16370 [math.PR]

arXiv 2025

[2] [2]

Entropic optimal transport: Geometry and large deviations.Duke Mathematical Journal, 171(16):3363–3400, 2022

Espen Bernton, Promit Ghosal, and Marcel Nutz. Entropic optimal transport: Geometry and large deviations.Duke Mathematical Journal, 171(16):3363–3400, 2022

2022

[3] [3]

Entropic approximation of∞-optimal transport problems.Applied Mathematics & Optimization, 90:10, 2024

Camilla Brizzi, Guillaume Carlier, and Luigi De Pascale. Entropic approximation of∞-optimal transport problems.Applied Mathematics & Optimization, 90:10, 2024

2024

[4] [4]

Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

Guillaume Carlier, Lénaïc Chizat, and Maxime Laborde. Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

2024

[5] [5]

Convergence of entropic schemes for optimal transport and gradient flows.SIAM Journal on Mathematical Analysis, 49(2):1385– 1418, 2017

Guillaume Carlier, Vincent Duval, Gabriel Peyré, and Bernhard Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows.SIAM Journal on Mathematical Analysis, 49(2):1385– 1418, 2017

2017

[6] [6]

Convergence rate of general entropic optimal transport costs.Calculus of Variations and Partial Differential Equations, 62:116, 2023

Guillaume Carlier, Paul Pegon, and Luca Tamanini. Convergence rate of general entropic optimal transport costs.Calculus of Variations and Partial Differential Equations, 62:116, 2023

2023

[7] [7]

On the zero-temperature limit of Gibbs states.Communic- ations in Mathematical Physics, 297:265–281, 2010

Jean-René Chazottes and Michael Hochman. On the zero-temperature limit of Gibbs states.Communic- ations in Mathematical Physics, 297:265–281, 2010

2010

[8] [8]

Regularity properties of the Schrödinger cost.ESAIM: Control, Optimisation and Calculus of Variations, 28:48, 2022

Gauthier Clerc. Regularity properties of the Schrödinger cost.ESAIM: Control, Optimisation and Calculus of Variations, 28:48, 2022

2022