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The convexity gap between row-averaged KL divergence and D_KL(μP||π) equals the mutual information I_μ(X;Y) for finite Markov chains.

2026-06-26 03:37 UTC pith:H7M7EWS4

load-bearing objection The paper gives a row-wise view of KL contraction via retention profiles, an exact mutual-information identity for the convexity gap, and an explicit construction separating localization from normalized contraction.

arxiv 2606.27073 v1 pith:H7M7EWS4 submitted 2026-06-25 math.PR

Retention Profiles and KL Contraction Bounds in Finite Markov Chains

Saurav Jadhav This is my paper
classification math.PR
keywords finite Markov chainsKL contractionretention profilemutual informationlocalization ratioSDPImixing timeCheeger bound
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Finite Markov chains admit a state-indexed retention profile r(x) obtained by normalizing each row's KL divergence to the stationary distribution π. A convexity-gap identity equates the difference between the π-averaged row divergences and the KL divergence of the pushed-forward measure to the mutual information between consecutive states under the initial distribution. This identity produces a decomposition of the KL contraction ratio into an entropy-inflation component and a mutual-information penalty. The work supplies an explicit sequence of chains demonstrating that the localization ratio L(P) can tend to zero without forcing the normalized contraction coefficient to approach one. It further records that L(P) equals one for every vertex-transitive chain and is uncorrelated with the spectral gap, Cheeger constant, and mixing time on the tested families.

Core claim

For a finite Markov chain with unique stationary distribution π, the retention profile is defined by r(x) = D_KL(P(x,·)||π) / log(1/π(x)), the maximum retention is M = max r(x), and the localization ratio is L(P) = E_π[r]/M. The convexity-gap identity states that the difference between the row-averaged divergence and D_KL(μP||π) equals I_μ(X;Y). The resulting decomposition expresses the contraction ratio as an entropy term minus a mutual-information penalty. An explicit construction of chains P_n shows that L(P_n) → 0 does not force η_KL(P_n)/M_n → 1, with the number of high-retention states being the decisive factor rather than their total π-mass.

What carries the argument

The retention profile r(x) together with the convexity-gap identity that sets the gap between averaged row KL divergences and D_KL(μP||π) equal to the mutual information I_μ(X;Y).

Load-bearing premise

The Markov chain is finite, possesses a unique stationary distribution π, and every row KL divergence to π is finite.

What would settle it

Direct numerical verification on the paper's explicit sequence of chains P_n: compute L(P_n) and η_KL(P_n)/M_n for increasing n and check whether the ratio approaches 1 whenever L approaches 0.

If this is right

  • The contraction ratio decomposes into entropy inflation minus a mutual-information penalty.
  • A Cheeger-type inequality supplies a lower bound on the maximum retention M in terms of the chain's bottleneck geometry.
  • Every vertex-transitive chain satisfies L(P) = 1 independently of its mixing speed.
  • L(P) is structurally independent of the spectral gap, Cheeger constant, and mixing time.
  • The retention profile tensorizes over product chains.

Where Pith is reading between the lines

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

  • The decoupling of L(P) from classical invariants indicates that retention profiles track contraction features orthogonal to standard mixing metrics.
  • The counterexample construction implies that the cardinality of high-retention states, rather than their aggregate mass, governs whether localization controls the normalized contraction coefficient.
  • The same gap identity may yield sharper tail bounds or reverse-Markov inequalities once the finite-state restriction is relaxed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript studies KL contraction in finite Markov chains from a row-wise perspective. It defines the retention profile r(x)=D_KL(P(x,·)||π)/log(1/π(x)) and the localization ratio L(P)=E_π[r]/M (with M=max r(x)). The central claims are (i) the exact convexity-gap identity E_μ[D(P(x,·)||π)]−D(μP||π)=I_μ(X;Y) together with a decomposition of the contraction ratio into entropy inflation and mutual-information penalty; (ii) a Cheeger-type lower bound on M; (iii) an explicit construction showing L(P_n)→0 need not imply η_KL(P_n)/M_n→1; and (iv) structural results including optimal tail bounds, Bhatia-Davis variance bound, two-sided spectral bounds with cubic correction, KL/Pinsker mixing-time bound, tensorization, and the fact that L(P) is decoupled from the spectral gap, Cheeger constant and mixing time (every vertex-transitive chain has L(P)=1 independently of mixing speed).

Significance. If the identities and explicit construction hold, the work supplies a new row-wise lens on KL contraction that cleanly separates localized versus global obstructions via the retention profile and localization ratio. The exact convexity-gap identity (derived from the joint KL expansion) and the counter-example construction are concrete strengths; the decoupling from classical mixing invariants on vertex-transitive chains is also noteworthy. These tools could usefully complement existing contraction-coefficient analyses in information theory and Markov-chain theory.

minor comments (3)
  1. Abstract states that the numerical experiments are exploratory and not used as evidence for any universal classification; repeat this disclaimer explicitly in the main text (e.g., near the description of the test suite) to prevent misreading.
  2. [Introduction] The definition of the SDPI ratio η_KL(P) is used throughout but is not restated in the introduction; add a one-sentence reminder of its standard definition for readers who may not recall the precise normalization.
  3. Notation for the averaged retention ar r_π and the maximum M is introduced in the abstract; ensure both symbols are defined at first use in the body and that the interval [0,1] for L(P) is justified immediately after the definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of the contributions, and recommendation of minor revision. The referee's description of the retention profile, localization ratio, convexity-gap identity, and decoupling results matches our manuscript closely. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core derivations begin from the definitions of the retention profile r(x) and localization ratio L(P) and apply standard probability identities such as the joint KL expansion D(joint || μ ⊗ π) to obtain the exact convexity-gap identity E_μ[D(P(x,·)||π)] − D(μP||π) = I_μ(X;Y). The Cheeger-type bound on M follows from bottleneck geometry applied to the row-retention profile, and the counterexample that L(P_n)→0 need not imply η_KL(P_n)/M_n→1 is supplied by explicit construction. Vertex-transitive symmetry forcing L(P)=1 is a direct algebraic consequence of the definition of r(x) being constant. No fitted parameters are renamed as predictions, no load-bearing self-citations appear, and the exploratory numerics are explicitly disclaimed as evidence for classification theorems. All steps remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper rests on standard assumptions for finite Markov chains and introduces new defined quantities rather than fitted parameters or postulated physical entities.

axioms (1)
  • domain assumption Finite Markov chain admits a unique stationary distribution π
    Required for all definitions involving π and KL divergences in the abstract.
invented entities (2)
  • Retention profile r(x) no independent evidence
    purpose: State-indexed measure of KL retention normalized by log(1/π(x))
    New definition introduced to enable row-wise analysis.
  • Localization ratio L(P) no independent evidence
    purpose: Ratio of average to max retention to distinguish contraction types
    New derived quantity L(P)=r_bar_π / M

pith-pipeline@v0.9.1-grok · 5891 in / 1502 out tokens · 66475 ms · 2026-06-26T03:37:44.540545+00:00 · methodology

0 comments
read the original abstract

We study Kullback-Leibler (KL) contraction in finite Markov chains through a row-wise perspective. Evaluating the SDPI ratio at point masses yields a state-indexed retention profile $r(x)=D_{\mathrm{KL}}(P(x,\cdot)\|\pi)/\log(1/\pi(x))$ and a localization ratio $L(P)=\bar r_\pi/M\in[0,1]$ (with $M=\max_x r(x)$, $\bar r_\pi=\mathbb{E}_\pi r$) that distinguishes localized from global contraction obstructions. Our main contributions are (i) a convexity-gap identity showing that the gap between the row-averaged divergence and $D_{\mathrm{KL}}(\mu P\|\pi)$ equals the mutual information $I_\mu(X;Y)$, and a derived decomposition of the contraction ratio into entropy inflation and a mutual-information penalty; (ii) a Cheeger-type lower bound on $M$, tying the bottleneck geometry of $P$ directly to the row-retention profile; (iii) an explicit construction proving that $L(P_n)\to 0$ does not force $\eta_{\mathrm{KL}}(P_n)/M_n\to 1$, identifying cardinality of high-retention states (not their $\pi$-mass) as the decisive quantity. Alongside these, we record structural consequences, optimal Markov/reverse-Markov tail bounds for $r$, a Bhatia-Davis variance bound, two-sided spectral bounds with an explicit cubic correction, a KL/Pinsker mixing-time bound, and tensorization for product chains. We further show that $L(P)$ is structurally decoupled from the spectral gap, the Cheeger constant, and the mixing time: every vertex-transitive chain satisfies $L(P)=1$ regardless of its mixing speed, and the empirical rank correlations between $L(P)$ and these classical invariants on a diverse but limited test suite are essentially zero. The numerical experiments are exploratory and not used as evidence for a universal classification theorem.

discussion (0)

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