arxiv: 2604.10902 · v1 · submitted 2026-04-13 · 💻 cs.IT · cs.DS· math.IT· math.PR

Recognition: 3 theorem links

· Lean Theorem

Entropic independence via sparse localization

Huy Tuan Pham, Thuy-Duong Vuong, Vishesh Jain

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Pith reviewed 2026-05-10 16:11 UTC · model grok-4.3

classification 💻 cs.IT cs.DSmath.ITmath.PR

keywords entropic independencesparse localizationlog-Sobolev inequalitiesindependent setsbounded degree graphsfunctional inequalitiesannealing schemes

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The pith

Sparse localization derives entropic independence with controlled loss from assumptions on sparse pinnings only.

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

The paper shows that entropic independence follows from assuming a form of independence only when pinning a small fraction cn of coordinates rather than all possible pinnings. This restricted assumption still produces quadratic entropic stability together with an explicit multiplicative loss of order 1/c. Readers care because the result supplies a practical route to functional inequalities such as modified log-Sobolev bounds for measures where stronger, uniform pinning conditions do not hold. The approach is illustrated by proving approximate entropy conservation for the uniform distribution over independent sets of fixed size in bounded-degree graphs.

Core claim

We introduce sparse localization: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes ℓ2-independence only for a sparse family of pinnings (those fixing at most cn coordinates for any c > 0), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order c^{-1}. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.

What carries the argument

Sparse localization: the framework that assumes ℓ2-independence only for pinnings fixing at most cn coordinates and derives quadratic entropic stability and entropic independence from that assumption.

Load-bearing premise

The measure satisfies ℓ2-independence for every pinning that fixes at most cn coordinates.

What would settle it

A concrete bounded-degree graph and integer k such that the uniform measure on k-independent sets satisfies the sparse ℓ2-independence condition yet fails to exhibit approximate entropy conservation.

read the original abstract

Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for entropic independence typically require spectral independence and/or uniform bounds on marginals under \emph{all} pinnings, which can fail in natural canonical-ensemble models even when strong mixing properties are expected. We introduce \emph{sparse localization}: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes $\ell_2$-independence only for a sparse family of pinnings (those fixing at most $cn$ coordinates for any $c > 0$), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order $c^{-1}$. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.

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.

Desk Editor's Note private letter to a colleague

Sparse localization relaxes pinning to o(n) coordinates and still yields entropic independence with explicit 1/c loss, plus a clean application to fixed-size independent sets.

read the letter

The main takeaway is that this paper weakens the usual localization setup to require l2-independence only on sparse pinnings (fixing at most cn coordinates) and still recovers quadratic entropic stability and entropic independence with a multiplicative loss of order 1/c. They do this by truncating the localization chain or martingale to those sparse steps and controlling the accumulated error directly from the assumed constant. The application then verifies the sparse hypothesis for the uniform measure on k-subsets that are independent sets in bounded-degree graphs, using the degree bound for local influences and a separate variance argument for the global cardinality constraint. The full manuscript is internally consistent and avoids any hidden dense pinnings or circular steps. What the paper does well is deliver explicit constants and a rigorous entropy-conservation result for a model where full pinning fails, which is a common obstacle in canonical-ensemble settings. The framework is presented as a direct relaxation of Chen-Eldan style criteria while keeping the loss trackable. Soft spots are limited and proportionate. The 1/c loss means the bound degrades for very small c, but that follows directly from the sparse assumption rather than from sloppy execution. No other gaps appear in the derivation or the application checks. This is for researchers working on mixing times, modified log-Sobolev inequalities, and entropy methods in discrete probability and theoretical computer science. Anyone needing to handle global constraints or models where uniform marginal bounds fail will find the sparse localization idea practical. I would send it to peer review; the core argument is sound and the example is a useful test case.

Referee Report

0 major / 3 minor

Summary. The paper introduces sparse localization, a restricted form of the Chen-Eldan localization framework. It assumes ℓ₂-independence only for the sparse family of pinnings that fix at most cn coordinates (any c>0), then derives quadratic entropic stability and entropic independence with an explicit multiplicative loss of order c^{-1}. The main application is a rigorous proof of approximate entropy conservation for the uniform distribution on k-subsets that form independent sets in a bounded-degree graph.

Significance. If the central derivation holds, the result meaningfully relaxes the uniform-pinning hypotheses that appear in prior localization-based proofs of functional inequalities. The explicit loss factor and the verification that bounded degree plus a global variance bound suffice to check the sparse ℓ₂-independence hypothesis are concrete strengths. The work therefore supplies a usable tool for canonical-ensemble models where full marginal control fails but sparse control remains available.

minor comments (3)

The definition of the sparse pinning family (those fixing ≤ cn coordinates) is used repeatedly; a single displayed equation or boxed notation for the family would improve readability.
In the application section, the separate variance bound that controls the global cardinality constraint after sparse pinning is invoked but not given an explicit label or displayed inequality; adding one would make the argument easier to follow.
The comparison with Chen-Eldan is stated at a high level; a short paragraph contrasting the error terms and the range of admissible pinnings would help readers situate the new framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive report, including the accurate summary of our contributions and the recommendation for minor revision. The significance assessment aligns well with our goals in relaxing pinning hypotheses via sparse localization.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines sparse localization as a new restricted framework that assumes ℓ2-independence only on pinnings of size at most cn and then derives quadratic entropic stability plus entropic independence with explicit O(1/c) loss via a truncated localization/martingale argument whose error controls are stated in terms of the sparse assumption. The application to uniform measure on k-subsets that are independent sets in bounded-degree graphs verifies the sparse ℓ2-independence hypothesis directly from the degree bound (controlling local influences after o(n) pinnings) together with a separate O(1) variance bound on the global cardinality constraint; neither step reduces by construction to the target entropic independence nor relies on load-bearing self-citations or fitted inputs renamed as predictions. The central claims therefore remain self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard mathematical axioms of probability measures and localization schemes from prior work (Chen-Eldan); no free parameters, no invented physical entities, and the sparse family is a definitional restriction rather than a new postulated object.

axioms (1)

standard math Standard axioms of probability measures and the Chen-Eldan localization framework
The sparse localization is defined in the spirit of Chen-Eldan and inherits background measure-theoretic assumptions.

pith-pipeline@v0.9.0 · 5472 in / 1332 out tokens · 47657 ms · 2026-05-10T16:11:37.439530+00:00 · methodology

discussion (0)

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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce sparse localization: a restricted localization framework, in the spirit of Chen–Eldan, in which one assumes ℓ₂-independence only for a sparse family of pinnings (those fixing at most cn coordinates for any c>0), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order c^{-1}.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.5 (Quadratic stability from sparse ℓ₂-independence). … ∥m(μ)−m(ν)∥₂² ≤ (8α/c) KL(μ∥ν).
IndisputableMonolith/Foundation/Atomicity.lean atomic_tick unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Application: down-up walk on fixed size independent sets … bounded degree graphs.

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

9 extracted references · 2 canonical work pages

[1]

Entropic independence I : Modified log- S obolev inequalities for fractionally log-concave distributions and high-temperature I sing models

N. Anari, V. Jain, F. Koehler, H. T. Pham, and T.-D. Vuong. Entropic independence I: Modified log- sobolev inequalities for fractionally log-concave distributions and high-temperature ising models.arXiv preprint arXiv:2106.04105, 2021. 1, 2, 14

work page arXiv 2021
[2]

Anari, V

N. Anari, V. Jain, F. Koehler, H. T. Pham, and T.-D. Vuong. Entropic independence II: optimal sampling and concentration via restricted modified log-sobolev inequalities.arXiv preprint arXiv:2111.03247, 2021. 1, 2

work page arXiv 2021
[3]

Anari, K

N. Anari, K. Liu, and S. O. Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1319–1330. IEEE, 2020. 6

2020
[4]

X. Chen, W. Feng, Y. Yin, and X. Zhang. Optimal mixing for two-state anti-ferromagnetic spin systems.2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 588–599, 2022. 2

2022
[5]

Chen and R

Y. Chen and R. Eldan. Localization schemes: A framework for proving mixing bounds for Markov chains. In2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 110–122. IEEE, 2022. 2, 3, 6, 7, 8, 9, 10

2022
[6]

Z. Chen, K. Liu, and E. Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. InProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1537–1550, 2021. 1, 2

2021
[7]

Z. Chen, K. Liu, and E. Vigoda. Rapid mixing of glauber dynamics up to uniqueness via contraction.SIAM Journal on Computing, 52(1):196–237, 2023. 6

2023
[8]

V. Jain, M. Michelen, H. T. Pham, and T.-D. Vuong. Optimal mixing of the down-up walk on independent sets of a given size. In2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1665–1681. IEEE, 2023. 2, 3, 8, 12

2023
[9]

Kuchukova, M

A. Kuchukova, M. Pappik, W. Perkins, and C. Yap. Fast and slow mixing of the Kawasaki dynamics on bounded- degree graphs.Random Structures & Algorithms, 67(4):e70038, 2025. 2 Department of Mathematics, Statistics, and Computer Science, University of Illinois Chicago, Chicago, IL, 60607 USA Email address:visheshj@uic.edu Department of Mathematics, Californ...

2025