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arxiv: 2604.14061 · v1 · submitted 2026-04-15 · 💻 cs.IT · math.IT· math.PR· stat.ML

Two-Sided Bounds for Entropic Optimal Transport via a Rate-Distortion Integral

Jingbo Liu This is my paper

Pith reviewed 2026-05-10 11:47 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PRstat.ML
keywords entropic optimal transportrate-distortion functionmutual information constraintGaussian processmajorizing measure theoreminformation inequalitiesoptimal transport bounds
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The pith

The maximum expected inner product between a random vector and the standard normal under a mutual information constraint equals a truncated rate-distortion integral up to universal constants.

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

The paper establishes that the largest possible expected inner product of a random vector with a standard normal vector, when taken over couplings whose mutual information is bounded or regularized, is equivalent to a truncated integral of the rate-distortion function. This equivalence is two-sided and holds with multiplicative constants that are universal across distributions. A sympathetic reader would care because the result converts an optimization problem arising in entropic optimal transport into a quantity already studied in rate-distortion theory, thereby supplying explicit bounds that can be evaluated or approximated by existing methods. The argument proceeds by lifting the original coupling to a Gaussian process indexed by a random subset of the type class and then invoking the majorizing measure theorem.

Core claim

The maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem.

What carries the argument

A lifting technique that constructs a Gaussian process indexed by a random subset of the type class of the probability distribution, to which a form of the majorizing measure theorem is applied.

If this is right

  • Two-sided bounds follow for quantities that arise in entropic optimal transport.
  • The mutual-information-constrained inner product can be approximated or bounded using standard rate-distortion calculations.
  • The equivalence holds with the same constants for any finite alphabet and any distribution on it.
  • Regularized transport problems become accessible to information-theoretic tools that already compute rate-distortion functions.

Where Pith is reading between the lines

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

  • The same lifting argument could be adapted to bound other linear functionals of the coupling beyond the inner product with a Gaussian vector.
  • Numerical algorithms that compute rate-distortion functions might be repurposed to estimate entropic optimal transport values without solving the transport problem directly.
  • The technique suggests that similar Gaussian-process embeddings could unify additional information inequalities with distortion theory.

Load-bearing premise

The lifting technique that constructs a Gaussian process indexed by a random subset of the type class, together with the applicability of the majorizing measure theorem to that process, yields the stated two-sided bounds.

What would settle it

Direct numerical evaluation of the maximum expected inner product for a discrete uniform distribution on a growing finite alphabet, compared against the corresponding truncated rate-distortion integral, to check whether their ratio remains bounded by fixed universal constants independent of alphabet size.

read the original abstract

We show that the maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem.

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

0 major / 2 minor

Summary. The manuscript establishes two-sided bounds showing that the maximum expected inner product between a random vector and the standard normal vector, taken over all couplings subject to a mutual information constraint or regularization, is equivalent up to universal multiplicative constants to a truncated integral involving the rate-distortion function. The proof constructs a Gaussian process indexed by a random subset of the type class of the underlying distribution via a lifting technique and applies the majorizing measure theorem to obtain matching upper and lower bounds on the expected supremum of the process.

Significance. If the result holds, it furnishes a precise quantitative link between entropic optimal transport and rate-distortion theory, with the universal constants providing robustness independent of specific distributions. The lifting construction combined with majorizing measures is a technically strong approach that yields explicit two-sided bounds rather than one-sided estimates, which is a clear strength of the work.

minor comments (2)
  1. The abstract refers to 'a form of the majorizing measure theorem' without naming the precise version or reference; adding a short parenthetical citation would improve immediate clarity for readers.
  2. Notation for the truncated integral in the main statement could be introduced with a brief display equation in the introduction to make the equivalence more immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the recognition of its technical contributions, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external probabilistic tools

full rationale

The central claim equates a variational problem (max expected inner product under mutual information constraint) to a truncated rate-distortion integral via a lifting construction that produces a Gaussian process on a random subset of the type class, followed by an application of the majorizing measure theorem. Both the lifting and the majorizing measure theorem are invoked as standard, independent results from probability theory; the paper does not define them in terms of the target equivalence or fit parameters to the output. No self-citation is load-bearing for the uniqueness or the bounds, and no step renames a fitted quantity as a prediction. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard information-theoretic definitions (mutual information, rate-distortion function) and the majorizing measure theorem from probability theory; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Standard properties of mutual information and rate-distortion function hold for the distributions under consideration
    Invoked implicitly when equating the constrained optimization to the rate-distortion integral
  • domain assumption The majorizing measure theorem applies to the constructed Gaussian process indexed by random type-class subsets
    Central to obtaining the two-sided bounds from the lifting construction

pith-pipeline@v0.9.0 · 5367 in / 1289 out tokens · 27021 ms · 2026-05-10T11:47:50.337495+00:00 · methodology

discussion (0)

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Reference graph

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