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arxiv: 2604.23229 · v1 · submitted 2026-04-25 · 🧮 math.ST · stat.TH

Solidarity of Spectral Gaps for Component-Wise Markov Chains

Youngwoo Kwon , Galin Jones , Qian Qin This is my paper

Pith reviewed 2026-05-08 07:12 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords spectral gapsMarkov chain Monte CarloGibbs samplerdeterministic scanrandom scancomponent-wise updatessolidarity principleMALA
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The pith

A block-wise contraction condition ensures that random-scan and deterministic-scan component-wise Markov chains have spectral gaps that are either both positive or both zero, differing by at most polynomial factors in the number of blocks.

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

The paper establishes a solidarity principle for the L2 spectral gaps of component-wise MCMC algorithms including Gibbs samplers. Under a block-wise contraction condition on the updates, the random-scan and deterministic-scan versions either both converge or both fail to converge, with gaps differing by factors polynomial in the number of blocks. This linkage allows transfer of convergence results between scan types. The authors apply it to derive a dimension-polynomial spectral gap bound for a deterministic-scan conditional MALA algorithm targeting multivariate Gaussians, relating the rate to the precision matrix and block step sizes.

Core claim

Under this condition, the spectral gaps of the random-scan and deterministic-scan versions of the Gibbs and component-wise chains are either simultaneously positive or simultaneously zero. Moreover, the spectral gaps differ by at most polynomial factors in the number of blocks. As an application, a deterministic-scan conditional MALA for multivariate Gaussian targets is studied, yielding a spectral gap bound that is polynomial in dimension by combining the condition with known random-scan Gibbs bounds.

What carries the argument

The block-wise contraction condition for the component-wise updates, which links the convergence behavior across different scanning strategies by controlling contraction of the target distribution.

If this is right

  • Positive spectral gap results for random-scan Gibbs samplers immediately imply corresponding results for deterministic-scan versions up to polynomial factors.
  • Convergence analysis of deterministic-scan conditional MALA on Gaussians reduces to properties of the precision matrix and chosen block step sizes.
  • High-dimensional sampling problems become more tractable when bounds can be transferred between scan strategies without separate proofs.
  • The choice between random and deterministic scans need not affect the qualitative convergence behavior of component-wise chains.

Where Pith is reading between the lines

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

  • Deterministic-scan methods, often used in practice for their predictability, can inherit convergence guarantees from their random-scan counterparts once the contraction condition is verified.
  • The polynomial dependence suggests that in very high dimensions the practical difference between scans may remain modest for mixing times.
  • Similar solidarity arguments could be tested for other component-wise Metropolis-Hastings variants beyond Gibbs and MALA.

Load-bearing premise

The component-wise block updates satisfy a contraction condition that controls the L2 distance to the target.

What would settle it

An explicit target distribution and update rule satisfying the block-wise contraction condition for which one scan type has a positive L2 spectral gap while the other has a zero gap, or where the gaps differ by a factor super-polynomial in the number of blocks.

Figures

Figures reproduced from arXiv: 2604.23229 by Galin Jones, Qian Qin, Youngwoo Kwon.

Figure 1
Figure 1. Figure 1: The web of implications for spectral gaps. Dashed arrows indicate known results: view at source ↗
read the original abstract

Deterministic-scan and random-scan component-wise Markov chain Monte Carlo algorithms, such as Gibbs samplers and conditional Metropolis-Hastings, are popular approaches for sampling from multivariate distributions. A long-standing open question is to determine the conditions under which these algorithms have similar convergence rates. A block-wise contraction condition for the component-wise updates is used to establish a solidarity principle for the $L^2$ spectral gaps of the associated Markov chains. Specifically, under this condition, the spectral gaps of the random-scan and deterministic-scan versions of the Gibbs and component-wise chains are either simultaneously positive or simultaneously zero. Moreover, the spectral gaps differ by at most polynomial factors in the number of blocks. As an application of the general results, a deterministic-scan conditional Metropolis-adjusted Langevin algorithm (MALA) for multivariate Gaussian targets is studied. The block-wise contraction condition is combined with known spectral gap bounds for the random-scan Gibbs sampler to obtain a spectral gap bound that is polynomial in dimension. The result is used to clarify how the convergence rate of the conditional MALA depends on the precision matrix of the Gaussian target and the step sizes of the block-wise MALA updates.

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

2 major / 2 minor

Summary. The paper proves a solidarity principle for the L² spectral gaps of random-scan and deterministic-scan versions of component-wise Markov chains (Gibbs samplers and conditional Metropolis-Hastings) under an explicitly stated block-wise contraction condition on the component-wise updates. Under this condition the gaps are simultaneously positive or simultaneously zero and differ by at most polynomial factors in the number of blocks. The result is applied to a deterministic-scan conditional MALA targeting multivariate Gaussians, where the contraction condition is combined with known random-scan Gibbs spectral-gap bounds to produce a spectral-gap lower bound that is polynomial in dimension; the dependence of the rate on the precision matrix and on the block-wise step sizes is then characterized.

Significance. If the solidarity result holds, it supplies a general, reusable device for transferring spectral-gap information between scan orders in component-wise MCMC without having to derive separate bounds for each ordering. The Gaussian MALA application demonstrates that the device yields concrete, dimension-polynomial guarantees and clarifies the role of the precision matrix and step-size choice. The approach reuses existing random-scan bounds rather than re-deriving them, which is a methodological strength for this class of problems.

major comments (2)
  1. [§3] §3 (solidarity theorem): the precise statement of the block-wise contraction condition (presumably Eq. (3.1) or (3.2)) is load-bearing; it would be useful to see an explicit verification that the condition is satisfied by the conditional MALA updates on the Gaussian target, including the dependence on the chosen step sizes.
  2. [Application section] Application section (conditional MALA): the final spectral-gap bound is stated to be polynomial in dimension, but the degree of the polynomial is not made explicit; because the solidarity result only guarantees “at most polynomial factors,” the overall degree should be tracked through the composition with the known random-scan Gibbs bound.
minor comments (2)
  1. The abstract claims the gaps “differ by at most polynomial factors in the number of blocks”; the main text should state the explicit degree (or at least an upper bound on it) for the reader’s convenience.
  2. Notation for the random-scan versus deterministic-scan operators should be introduced once and used consistently; occasional switches between “RS”/“DS” and full phrases make some passages harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive comments. We address the two major comments point by point below and will incorporate revisions to improve clarity.

read point-by-point responses

  1. Referee: [§3] §3 (solidarity theorem): the precise statement of the block-wise contraction condition (presumably Eq. (3.1) or (3.2)) is load-bearing; it would be useful to see an explicit verification that the condition is satisfied by the conditional MALA updates on the Gaussian target, including the dependence on the chosen step sizes.

    Authors: We agree that an explicit verification strengthens the application. In the revised manuscript we will add a dedicated paragraph (or short appendix subsection) that directly verifies the block-wise contraction condition for the conditional MALA updates on the multivariate Gaussian target. The verification will be carried out component-wise, explicitly displaying the dependence on the chosen step sizes and on the eigenvalues of the precision matrix, confirming that the condition holds whenever the step sizes are chosen sufficiently small relative to the local curvature. revision: yes

  2. Referee: [Application section] Application section (conditional MALA): the final spectral-gap bound is stated to be polynomial in dimension, but the degree of the polynomial is not made explicit; because the solidarity result only guarantees “at most polynomial factors,” the overall degree should be tracked through the composition with the known random-scan Gibbs bound.

    Authors: We acknowledge the value of tracking the degree. The solidarity theorem already states that the factors are polynomial in the number of blocks (here equal to dimension d), with the precise exponent depending on the contraction constants appearing in the block-wise condition. Composing with the known random-scan Gibbs bound therefore yields an overall polynomial whose degree is the sum of the solidarity exponent and the degree of the random-scan bound. In the revision we will add an explicit remark after the statement of the final bound that records this additive structure and notes the dependence on the precision-matrix eigenvalues through the random-scan bound; we do not claim a universal numerical degree because it necessarily varies with the target covariance and step-size choice. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central result is a solidarity theorem for L2 spectral gaps of random-scan versus deterministic-scan component-wise chains (Gibbs and conditional MH), derived directly from an explicitly stated block-wise contraction condition on the component-wise updates. This condition functions as an assumption, not a fitted or self-derived quantity. The MALA application then combines the solidarity result with external, pre-existing spectral-gap bounds for random-scan Gibbs samplers on Gaussians; those bounds are cited as known and independent of the present work. No equation reduces by construction to its own inputs, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The derivation is therefore self-contained against the stated assumption and external literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the block-wise contraction condition (a domain assumption) together with standard background facts about Markov operators, L2 spectral gaps, and Gibbs samplers; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Block-wise contraction condition on component-wise updates
    This is the key hypothesis used to prove that random-scan and deterministic-scan spectral gaps are simultaneously positive or zero.

pith-pipeline@v0.9.0 · 5499 in / 1277 out tokens · 43440 ms · 2026-05-08T07:12:18.090999+00:00 · methodology

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    42 Then, for every𝑎∈ (0,1), Φ(𝑀 2 Δ 𝑗 ) ≥𝜖min 1−𝑎 2 , 𝑎2𝜅𝑡 4

    Three-set isoperimetric inequality:For any measurable partition{𝑆1, 𝑆2, 𝑆3} of R𝑁 𝑗 such thatd(𝑆 1, 𝑆2) ≥𝑡, 𝛾𝑁 𝑗 (𝑆 3) ≥𝜅𝑡𝛾 𝑁 𝑗 (𝑆 1)𝛾 𝑁 𝑗 (𝑆 2). 42 Then, for every𝑎∈ (0,1), Φ(𝑀 2 Δ 𝑗 ) ≥𝜖min 1−𝑎 2 , 𝑎2𝜅𝑡 4 . Since MALA consists of a Gaussian proposal followed by a Metropolis step, we first bound the total variation distance between the proposal distribut...

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  55. [55]

    For any𝑢, 𝑣∈R 𝑁 𝑗, supposedΔ 𝑗 (𝑢, 𝑣) ≤𝑡=𝜂 √

    Set 𝜖=𝜀(𝜂,Δ 𝑗 )= 2 1+𝛿 max exp ( − tr(Δ 2 𝑗 ) 2(1−𝛿 max) ) −2Φ 𝑁 ∥𝐼−Δ 𝑗 ∥op 2 𝜂 . For any𝑢, 𝑣∈R 𝑁 𝑗, supposedΔ 𝑗 (𝑢, 𝑣) ≤𝑡=𝜂 √

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    By assumption, tr(Δ 2 𝑗 )=ℎ 2 𝑗 tr(𝑄 2 𝑗, 𝑗 ) ≤ 1 10 ≤log(10/9), and since𝛿max ≤1/2, this implies tr(Δ 2 𝑗 ) ≤2(1−𝛿 max)log 5/3 1+𝛿 max

    Since 𝛿max =𝜓 max(Δ 𝑗 ) ≤1/2<1 , Lemma D.7 yields ∥𝑀 2 Δ 𝑗 (𝑢,·) −𝑀 2 Δ 𝑗 (𝑣,·) ∥ TV ≤ ∥𝑀 Δ 𝑗 (𝑢,·) −𝑀 Δ 𝑗 (𝑣,·) ∥ TV ≤1−𝜖 . By assumption, tr(Δ 2 𝑗 )=ℎ 2 𝑗 tr(𝑄 2 𝑗, 𝑗 ) ≤ 1 10 ≤log(10/9), and since𝛿max ≤1/2, this implies tr(Δ 2 𝑗 ) ≤2(1−𝛿 max)log 5/3 1+𝛿 max . Hence the first term in the definition of𝜖 is at least6/5. Also, since∥𝐼−Δ 𝑗 ∥op =1−𝜓 min(Δ 𝑗 ...

    work page

  57. [57]

    Hence Φ(𝑀 2 Δ 𝑗 ) ≥𝜖 𝜅𝑡 8 ≥ √︁ 𝜓min(Δ 𝑗 ) 80√𝜋

    Since 𝜓min(Δ 𝑗 ) ≤𝜓 max(Δ 𝑗 )=ℎ 𝑗 𝜓max(𝑄 𝑗, 𝑗 ) ≤ 1 2, we have𝜅𝑡= √︁ 𝜓min(Δ 𝑗 )/𝜋 <1 , which implies(1−𝑎 ★)/2≥1/8 and 𝑎2 ★𝜅𝑡/4=𝜅𝑡/8≤1/8 . Hence Φ(𝑀 2 Δ 𝑗 ) ≥𝜖 𝜅𝑡 8 ≥ √︁ 𝜓min(Δ 𝑗 ) 80√𝜋 . By Lemma D.3, 1− ∥𝑀 2 Δ 𝑗 −Π 𝛾 𝑗 ∥ ≥ Φ(𝑀 2 Δ 𝑗 )2 8 ≥ 𝜓min(Δ 𝑗 ) 51200𝜋 . Finally, by (18), 1− ∥𝑀 Δ 𝑗 −Π 𝛾 𝑗 ∥ ≥ 1− ∥𝑀 2 Δ 𝑗 −Π 𝛾 𝑗 ∥ 2 ≥ 𝜓min(Δ 𝑗 ) 102400𝜋 , 49 which yi...

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