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

Jingbo Liu

Apr 15, 2026arXiv:2604.14061v1

cs.ITmath.PRstat.ML

Scorecard· 5/16

7.5/10 impact

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.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

Core Contribution

This paper establishes two-sided (matching up to universal constants) bounds for entropic optimal transport — both the information-constrained version $w(\gamma, \mu, R)$ and the information-regularized version $f(\gamma, \mu, \beta)$ — in terms of truncated integrals involving the rate-distortion function. Specifically, Theorem 1 shows:

$w(\gamma, \mu, R) \asymp \int_0^\infty \sqrt{R \wedge i_\mu(\sigma)} \, d\sigma$

and Theorem 2 provides an analogous characterization for the regularized version in terms of $\varphi(\mu, \alpha) = \inf_{\sigma > 0}\{i_\mu(\sigma^2) + \sigma^2/\alpha^2\}$ .

The key innovation over the prior work [1] (which established the unconstrained case $R = \infty$ ) is the incorporation of the mutual information constraint/regularization, which requires a truncation of the integral. The truncation $R \wedge i_\mu(\sigma)$ elegantly captures the interplay between the information budget $R$ and the intrinsic geometric complexity of $μ \mu$ at scale $\sigma$ .

Methodological Rigor

The proof strategy is technically sophisticated, building on a "lifting" technique that constructs a Gaussian process indexed by a random subset of the type class. This is a meaningful refinement over [1], which used the full type class. The key steps are:

1. Lemma 2 establishes a representation connecting $w(\gamma, \mu, R)$ to the expected maximum of an inner product over a random subset of size $\lfloor e^{NR} \rfloor$ drawn from the type class. This is proved via careful first-moment (union bound) and second-moment arguments using the method of types.

2. Concentration analysis: The paper shows that the random subset approximately preserves the "stationarity-like" structure of the full type class. Specifically, the fraction of points in any ball concentrates with double-exponential tails (Lemma 3), ensuring that the empirical measure $\mu_N$ on the random subset approximates the type class geometry. The truncation $R \wedge i_\mu(\sigma)$ precisely restricts to the regime where this concentration holds.

3. Majorizing measure machinery: The lower bound uses $\delta_2(T)$ (the dual form of $\gamma_2$ ), while the upper bound leverages Dudley's integral. The connection between these and the rate-distortion integral is mediated by Lemma 4, a change-of-variables identity.

The arguments appear sound, though the paper is dense and some steps (e.g., the passage from rational to general measures, the limiting arguments) are sketched rather than fully detailed. The reliance on the equivalence $\delta_2 \asymp \gamma_2$ from Talagrand's theory means the universal constants are inherited and not explicitly computed.

Potential Impact

Theoretical significance: This result sits at a rich intersection of information theory, optimal transport, and the theory of Gaussian processes. The two-sided nature is important — one-sided bounds (like Dudley's integral) are significantly less powerful. The fact that Theorem 2 enjoys exact tensorization is particularly noteworthy, as tensorization properties are fundamental in high-dimensional probability and information theory. The paper raises the tantalizing question of whether this "incremental form" could lead to a purely analytic proof of the majorizing measure theorem.

Potential applications:

Machine learning: Entropic regularization is central to computational optimal transport (Sinkhorn's algorithm). These bounds could inform convergence rates or approximation guarantees.

Statistical estimation: The connection to generalization bounds (mentioned in the introduction via [1]) and statistical regression suggests concrete downstream applications.

Generative models: The Gaussian case is directly relevant to score-based diffusion models and other generative frameworks where transport from Gaussian to a target measure is fundamental.

Connections to prior art: The paper clearly contextualizes itself relative to [1], [8], [10], and [11]. The dimension-free nature of the bounds distinguishes this from [10]. Compared to [11] (Chu-Raginsky), which obtains one-sided bounds for finite index sets, this work achieves two-sided bounds in full generality.

Timeliness & Relevance

The paper addresses a current convergence of interest between optimal transport theory and information theory, fields that are increasingly intertwined in modern machine learning theory. Entropic regularization is arguably the most practically important variant of optimal transport, making theoretical characterizations of the entropic transport cost timely. The connection to the majorizing measure theorem — one of the deepest results in probability theory — adds fundamental mathematical interest.

Strengths

1. Two-sided bounds with universal constants: This is the gold standard in this area, and achieving it for the information-constrained/regularized setting is non-trivial.

2. Elegant structural insight: The truncated integral $\int \sqrt{R \wedge i_\mu(\sigma)} \, d\sigma$ provides a clean, interpretable characterization.

3. Exact tensorization of the regularized version (Theorem 2) is a powerful structural property.

4. The random subset technique is a novel methodological contribution that may find applications elsewhere.

5. Unifying perspective: The paper connects rate-distortion theory, Gaussian processes, and optimal transport in a coherent framework.

Limitations

1. Conference format constraints: Several arguments are compressed or deferred; full verification requires checking implicit claims carefully.

2. Universal constants are not explicit, inherited from the majorizing measure theorem.

3. Practical applicability remains speculative — the paper does not include computational examples or demonstrate how the bounds could be used algorithmically.

4. Restriction to Gaussian reference measure: While important and well-motivated, extending beyond $\gamma = N(0, I_n)$ would broaden applicability.

5. Comparison with [11] could be more detailed, particularly regarding the finite index set case where both papers apply.

Overall Assessment

This is a technically strong theoretical contribution that extends a beautiful recent result (the rate-distortion integral) to the practically important entropic setting. The methodology is creative, combining information-theoretic and probabilistic tools in a non-trivial way. While the paper is narrowly focused and primarily theoretical, the results are clean, the proof technique is novel, and the connections to multiple areas of mathematics and machine learning are compelling.

Rating:7.5/ 10

Significance 7.5Rigor 7.5Novelty 7Clarity 6.5

Generated Apr 16, 2026

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Scorecard· 5/16

7.5/10 impact