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arxiv: 2604.20827 · v1 · submitted 2026-04-22 · 🧮 math.PR · math.OC

Failure of ambient closed-set large-deviation upper bounds in entropic optimal transport

Maja Gwozdz This is my paper

Pith reviewed 2026-05-09 22:56 UTC · model grok-4.3

classification 🧮 math.PR math.OC
keywords entropic optimal transportlarge deviationsclosed setsupper boundsnonatomic marginalsfixed-cost modeltotal variation convergencetail criterion
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The pith

Large-deviation upper bounds valid on compact sets fail on closed sets in entropic optimal transport.

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

The paper constructs a fixed-cost model with continuous cost and nonatomic marginals in which entropic minimizers converge in total variation to an optimal plan whose support is noncompact. In this setting the standard large-deviation upper bound continues to hold on compact sets, yet the analogous bound fails on a chosen closed subset of the ambient space. The work supplies the exact tail criterion that must hold for the closed-set bound to be recovered from the compact-set version. It further shows that no full large-deviation principle exists on the whole space at speed 1/ε with an arbitrary lower-semicontinuous rate function. A sympathetic reader cares because the example demonstrates that an apparently technical obstruction already appears inside the core setting of static entropic optimal transport.

Core claim

We construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support. The known compact-set large-deviation upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set we identify the exact tail criterion for passing from compact to closed sets. There does not exist a full large-deviation principle on the ambient space at speed 1/ε with an arbitrary lower semicontinuous rate function.

What carries the argument

A fixed-cost model with continuous cost and nonatomic marginals that forces entropic minimizers to converge in total variation to a noncompactly supported optimal plan; this model serves as a counterexample showing that compact-set upper bounds do not extend automatically to closed sets.

If this is right

  • For any fixed closed set the closed-set upper bound holds precisely when a specific tail criterion is satisfied.
  • No full large-deviation principle exists on the ambient space at speed 1/ε with an arbitrary lower semicontinuous rate function.
  • Entropic minimizers can converge to noncompactly supported plans even under continuous costs and nonatomic marginals.
  • The obstruction to extending large-deviation bounds from compact to closed sets already appears in static entropic optimal transport.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Users of large-deviation bounds in transport problems must verify explicit tail conditions before invoking closed-set versions.
  • The same construction may be adapted to detect analogous failures in dynamic or multi-marginal entropic settings.
  • Rate functions arising in these problems may need built-in tail decay to guarantee closed-set validity.

Load-bearing premise

The existence of a fixed-cost model with continuous cost and nonatomic marginals such that entropic minimizers converge in total variation to a noncompactly supported optimal plan.

What would settle it

A calculation in the constructed model that shows the closed-set upper bound holds on the chosen closed subset, or that the limit plan actually has compact support.

read the original abstract

Large-deviation upper bounds on compact sets do not, in general, extend to arbitrary closed sets without additional tightness. We show that this obstruction already occurs in static entropic optimal transport. More precisely, we construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support, the known compact-set upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set, we identify the exact tail criterion for passing from compact to closed sets. We show that there does not exist a full large-deviation principle (LDP) on the ambient space at speed $1/\varepsilon$ with an arbitrary lower semicontinuous rate function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper shows that large-deviation upper bounds on compact sets for the entropic optimal transport problem do not extend in general to arbitrary closed sets in the ambient space without additional tightness. It constructs an explicit fixed-cost model with continuous cost and nonatomic marginals in which the entropic minimizers converge in total variation to an optimal plan whose support is noncompact; the known compact-set upper bound continues to hold, but the closed-set upper bound fails on a chosen closed subset. For any fixed closed set the authors identify the precise tail criterion separating the compact and closed cases, and they prove that no large-deviation principle holds on the whole ambient space at speed 1/ε with a lower-semicontinuous rate function.

Significance. The result supplies a concrete, parameter-free counterexample that isolates the exact obstruction (noncompact support of the limiting plan) preventing closed-set upper bounds in entropic OT. By exhibiting both the failure on a specific closed set and the tail criterion that restores the bound, the paper clarifies the boundary between compact and closed large-deviation statements in this setting. The explicit construction and the negative result on the existence of a full LDP are the main contributions.

major comments (2)
  1. [Construction and proof of the counterexample] The central construction (fixed-cost model, continuous cost, nonatomic marginals, TV convergence to noncompact-support plan) is load-bearing for every claim. The manuscript must supply a fully detailed verification that the chosen marginals are indeed nonatomic, that the cost is continuous, and that the tail behavior produces the stated violation of the closed-set upper bound on the chosen closed subset.
  2. [Non-existence of full LDP] The argument that no lsc rate function can yield a full LDP on the ambient space relies on the same noncompact-support limit plan. A self-contained proof that any candidate rate function would have to be infinite on the chosen closed set while remaining finite on compact sets should be given explicitly.
minor comments (2)
  1. Notation for the entropic functional, the speed 1/ε, and the topology on the space of measures should be introduced once and used consistently.
  2. The precise statement of the tail criterion (for a fixed closed set) would benefit from being isolated as a numbered proposition or corollary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Construction and proof of the counterexample] The central construction (fixed-cost model, continuous cost, nonatomic marginals, TV convergence to noncompact-support plan) is load-bearing for every claim. The manuscript must supply a fully detailed verification that the chosen marginals are indeed nonatomic, that the cost is continuous, and that the tail behavior produces the stated violation of the closed-set upper bound on the chosen closed subset.

    Authors: We agree that a fully detailed verification of the central construction is essential. In the revised manuscript we will expand the relevant section to include: (i) an explicit check that both marginals are nonatomic, by verifying that every singleton has measure zero; (ii) a direct verification that the cost function is continuous on the product space; and (iii) a step-by-step computation of the relevant tail probabilities that produces the violation of the closed-set upper bound on the chosen closed subset. These additions will render the counterexample self-contained. revision: yes

  2. Referee: [Non-existence of full LDP] The argument that no lsc rate function can yield a full LDP on the ambient space relies on the same noncompact-support limit plan. A self-contained proof that any candidate rate function would have to be infinite on the chosen closed set while remaining finite on compact sets should be given explicitly.

    Authors: We agree that the non-existence argument can be made more explicit. In the revision we will supply a self-contained proof in a dedicated subsection: assuming an LDP holds at speed 1/ε with a lower-semicontinuous rate function I, the known compact-set upper bounds imply that I is finite on every compact set; lower semicontinuity together with the total-variation convergence to the noncompact-support limit plan then forces I to be infinite on the chosen closed set, yielding the desired contradiction. This argument will be written out in full detail without relying on external references. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript establishes its main result via an explicit counterexample construction: a fixed-cost model with continuous cost and nonatomic marginals in which entropic minimizers converge in total variation to a noncompact-support optimal plan, compact-set upper bounds hold, and a chosen closed set violates the closed-set upper bound. This is a direct, self-contained existence argument that does not reduce any claimed prediction or rate function to a fitted parameter, self-definition, or prior self-citation chain. The tail criterion separating compact from closed sets and the non-existence of a full LDP are derived from the constructed model itself rather than imported via ansatz or uniqueness theorem. No load-bearing step collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a particular fixed-cost model with continuous cost and nonatomic marginals whose entropic minimizers converge to a noncompact-support plan; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of large deviation upper bounds on compact sets and lower semicontinuity of rate functions hold in the ambient space.
    Invoked when stating that the compact-set upper bound remains valid while the closed-set version fails.
  • domain assumption Convergence in total variation of entropic minimizers to an optimal plan is well-defined for the chosen model.
    Used to link the limit behavior to the failure of the closed-set bound.

pith-pipeline@v0.9.0 · 5427 in / 1341 out tokens · 24256 ms · 2026-05-09T22:56:25.689033+00:00 · methodology

discussion (0)

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Reference graph

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