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arxiv: 2604.05861 · v1 · submitted 2026-04-07 · 🧮 math.PR

Quantitative bounds for high dimensional entropic CLT

Chang-song Deng , Lin Wang , Lihu Xu This is my paper

Pith reviewed 2026-05-10 18:46 UTC · model grok-4.3

classification 🧮 math.PR
keywords entropic central limit theoremhigh dimensionsPoincaré inequalityHarnack inequalityprojection methodquantitative boundsconvergence rates
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The pith

Extending the Johnson-Barron projection method to high dimensions yields new quantitative bounds for the entropic central limit theorem under the Poincaré inequality.

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

The paper is trying to establish a new quantitative bound on how quickly the entropy of normalized sums of independent high-dimensional random vectors approaches the entropy of the limiting Gaussian. It does this by taking the Johnson-Barron projection method, which works in one dimension, and extending it to higher dimensions with the help of a special inequality that does not depend on the dimension. A reader would care because better bounds in high dimensions can improve approximations used in statistics and physics where many variables interact. The authors also compare their bound to other recent results to highlight its strengths.

Core claim

By extending the Johnson--Barron projection method from one dimension to high dimensions and utilizing a Wang type dimension-free Harnack inequality, a new quantitative bound is obtained for the entropic central limit theorem under the assumption that the Poincaré inequality holds.

What carries the argument

The high-dimensional extension of the Johnson--Barron projection method, which works together with a dimension-free Harnack inequality to produce the entropy convergence bound.

If this is right

  • The new bound applies directly in high dimensions where one-dimensional methods do not extend automatically.
  • Comparison with recent results demonstrates concrete advantages of the projection-plus-Harnack approach.
  • The bound remains useful even as dimension grows, provided the Poincaré inequality is satisfied.
  • Quantitative rates become available for entropy convergence in vector-valued central limit settings.

Where Pith is reading between the lines

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

  • The same extension strategy might produce similar bounds for other functional inequalities such as logarithmic Sobolev.
  • Numerical checks on product distributions or uniform measures on high-dimensional balls could quickly test whether the rates match observed entropy decay.
  • The dimension-free control may transfer to non-iid sums or to settings with weak dependence.

Load-bearing premise

The underlying distributions satisfy the Poincaré inequality.

What would settle it

A concrete high-dimensional distribution that obeys the Poincaré inequality but for which the entropy distance to the Gaussian exceeds the paper's stated quantitative bound.

read the original abstract

By extending the Johnson--Barron projection method from one dimension to high dimensions and utilizing a Wang type dimension-free Harnack inequality, we obtain a new quantitative bound for the entropic central limit theorem under the assumption that the Poincar\'e inequality holds. We compare our results with recent developments to demonstrate the merits of our approach.

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 / 3 minor

Summary. The manuscript extends the Johnson--Barron projection method from one to high dimensions and combines it with a Wang-type dimension-free Harnack inequality to derive a quantitative bound for the entropic central limit theorem, conditional on the Poincaré inequality holding for the underlying distributions. The bound is compared to recent results to illustrate its merits.

Significance. If the derivation is correct, the result supplies a new quantitative entropic CLT bound that remains dimension-free under the standard Poincaré assumption. The explicit combination of projection techniques with a dimension-free Harnack inequality is a clear technical contribution and could be useful for further high-dimensional quantitative limit theorems.

minor comments (3)
  1. The abstract states that a derivation exists but does not indicate the precise form of the quantitative bound (e.g., dependence on dimension, entropy deficit, or constants). Adding a brief display of the main inequality in the abstract or introduction would improve readability.
  2. Section 2 (or wherever the high-dimensional projection is defined) should explicitly state how the one-dimensional Johnson--Barron argument is lifted, including any new error terms introduced by the extension.
  3. The comparison with recent developments would be strengthened by a short table listing the assumptions, dimension dependence, and rate of the new bound versus the cited works.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we interpret this as an invitation to make any necessary minor clarifications or corrections in the revised version while preserving the core contribution.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation extends the Johnson-Barron projection method to high dimensions and combines it with a Wang-type dimension-free Harnack inequality to produce a quantitative entropic CLT bound, but only under the explicit external assumption that the Poincaré inequality holds for the underlying distributions. No equations or steps in the provided abstract or description reduce the target bound to a fitted parameter, a self-referential definition, or a load-bearing self-citation chain. The result is presented as conditional on an independent assumption and built from known techniques, so the central claim remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Poincaré inequality as a domain assumption and on the validity of the extended projection method and Harnack inequality; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Poincaré inequality holds for the distributions
    Explicitly required in the abstract to obtain the quantitative bound for the entropic CLT.

pith-pipeline@v0.9.0 · 5334 in / 1150 out tokens · 37959 ms · 2026-05-10T18:46:27.788744+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    By extending the Johnson–Barron projection method from one dimension to high dimensions and utilizing a Wang type dimension-free Harnack inequality, we obtain a new quantitative bound for the entropic central limit theorem under the assumption that the Poincaré inequality holds.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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