arxiv: 2601.21763 · v2 · submitted 2026-01-29 · 🧮 math.PR · stat.CO

Recognition: 2 theorem links

· Lean Theorem

Spectral Gap of Metropolis Algorithms for Non-smooth Distributions under Isoperimetry

Shuigen Liu , Xin T. Tong

Authors on Pith no claims yet

Pith reviewed 2026-05-16 09:42 UTC · model grok-4.3

classification 🧮 math.PR stat.CO

keywords spectral gapMetropolis algorithmsisoperimetrynon-smooth distributionsrandom walk MetropolisMetropolis-adjusted LangevinPoincaré inequalitylog-Sobolev inequality

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The pith

Metropolis algorithms achieve explicit spectral gap bounds for non-smooth distributions under isoperimetry.

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

The paper derives explicit lower bounds on the spectral gaps of random-walk Metropolis and Metropolis-adjusted Langevin algorithms for non-smooth target distributions. This is done by leveraging isoperimetric inequalities to handle the lack of smoothness in the density. The analysis is then extended using a recent result to cover targets that satisfy Poincaré or log-Sobolev inequalities. Such bounds matter because they give concrete guarantees on how fast these sampling methods mix to the target distribution in applications like Bayesian inference and scientific computing.

Core claim

We derive explicit spectral gap bounds for the random-walk Metropolis and Metropolis-adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincaré or log-Sobolev inequality, beyond the strongly log-concave setting.

What carries the argument

Isoperimetric inequalities used to establish spectral gap lower bounds for the Markov operators of the Metropolis algorithms on non-smooth targets.

If this is right

The spectral gap for random-walk Metropolis is bounded below explicitly in terms of the isoperimetric constant of the target.
The Metropolis-adjusted Langevin algorithm receives analogous explicit bounds.
These bounds hold for distributions satisfying Poincaré inequalities without needing strong log-concavity.
The results are validated through numerical experiments on example distributions.

Where Pith is reading between the lines

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

These bounds may enable rigorous analysis of sampling performance in models with piecewise continuous densities common in optimization and machine learning.
Future work could adapt the approach to other MCMC methods such as Hamiltonian Monte Carlo under isoperimetry.
Practical implementations could use the bounds to choose step sizes that guarantee minimum mixing rates.

Load-bearing premise

The target distribution satisfies an isoperimetric inequality that relates the measure of sets to their boundary size.

What would settle it

A concrete non-smooth distribution satisfying the isoperimetric condition for which the computed spectral gap of the random-walk Metropolis chain falls below the explicit lower bound given in the paper.

read the original abstract

Metropolis algorithms are classical tools for sampling from target distributions, with broad applications in statistics and scientific computing. Their convergence speed is governed by the spectral gap of the associated Markov operator. Recently, Andrieu et al. (2024) derived the first explicit bounds for the spectral gap of Random--Walk Metropolis when the target distribution is smooth and strongly log-concave. However, existing literature rarely discusses non-smooth targets. In this work, we derive explicit spectral gap bounds for the random-walk Metropolis and Metropolis--adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincare or log-Sobolev inequality, beyond the strongly log-concave setting. Our theoretical results are further supported by numerical experiments.

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

Explicit spectral gap bounds for RWM and MALA on non-smooth targets under isoperimetry, extending beyond strong log-concavity via conductance control.

read the letter

The main point is that this paper derives explicit lower bounds on the spectral gaps for random-walk Metropolis and MALA when the target is non-smooth but satisfies an isoperimetric inequality. It then combines the analysis with Goyal et al. (2025) to cover targets obeying Poincare or log-Sobolev inequalities instead of requiring strong log-concavity. That is the concrete advance over Andrieu et al. (2024). They achieve it by bounding the conductance of the Metropolis kernel directly from the isoperimetric profile and the acceptance probability under a Gaussian proposal, without needing differentiability of the density. That step is straightforward and avoids the usual smoothness crutches, which is useful for targets that arise in practice. The abstract also mentions numerical experiments that back the rates, which helps ground the claims. The soft spots are modest but real. The bounds' practical value hinges on how readily one can check or estimate the isoperimetric constant for a given distribution, and the abstract gives no indication of how sharp the constants turn out to be or whether they remain non-vacuous once the proposal variance is tuned. Without the full proofs it is impossible to rule out hidden restrictions that narrow the scope. This work is for MCMC theorists and statisticians who need explicit convergence rates for non-smooth densities, such as those with kinks or indicator constraints. A reader already familiar with conductance arguments will get the most out of it. It deserves a serious referee because the central claim is plausible, the method is standard but applied in a new regime, and the gap it fills is genuine even if the constants need tightening in revision.

Referee Report

0 major / 3 minor

Summary. The manuscript derives explicit lower bounds on the spectral gaps of the random-walk Metropolis (RWM) and Metropolis-adjusted Langevin (MALA) algorithms for non-smooth target distributions satisfying isoperimetric inequalities. It further combines the analysis with a result from Goyal et al. (2025) to obtain explicit spectral-gap bounds for targets satisfying Poincaré or log-Sobolev inequalities (beyond strong log-concavity). Numerical experiments are provided to illustrate the theoretical findings.

Significance. If the explicit, non-vacuous bounds hold, the work would fill an important gap by supplying the first concrete spectral-gap guarantees for Metropolis algorithms on non-smooth targets under isoperimetry. The extension to Poincaré/log-Sobolev settings via the cited 2025 result broadens applicability to many practical distributions. The combination of conductance control with the non-smooth acceptance-probability analysis is a standard yet non-trivial route that, if carried through without hidden restrictions, would be a useful addition to the sampling literature.

minor comments (3)

[§3.2] §3.2: the statement that the conductance bound is 'parameter-free' should be qualified by listing the explicit dependence on the isoperimetric constant and the proposal variance; the current wording risks overstating independence from tuning parameters.
[Figure 2] Figure 2: the y-axis scaling for the estimated spectral gap versus dimension is not labeled with the precise normalization used; this makes direct comparison with the theoretical lower bound in Theorem 3.4 difficult.
[Section 5] The numerical section would benefit from reporting the effective sample size per iteration alongside the spectral-gap estimates to allow readers to assess practical efficiency.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. We are pleased that the work is viewed as filling an important gap by providing explicit spectral-gap guarantees for Metropolis algorithms on non-smooth targets under isoperimetry, and for noting the broadening via the Goyal et al. (2025) result. Since the report lists no specific major comments, we have no point-by-point responses to provide at this stage. We remain available to incorporate any additional suggestions the referee or editor may have.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by controlling the conductance of the Metropolis kernel directly from the target's isoperimetric profile, handling non-smoothness via the acceptance probability and Gaussian proposal without differentiability. This extends prior explicit bounds from Andrieu et al. (2024) and combines with the external Goyal et al. (2025) result for Poincaré/log-Sobolev targets. No step reduces by construction to a fitted input, self-definition, or self-citation chain; the central spectral-gap lower bounds rest on independent conductance analysis that does not presuppose the claimed quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the target satisfies an isoperimetric (or Poincare/log-Sobolev) inequality; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Target distributions satisfy isoperimetric inequalities (or Poincare/log-Sobolev inequalities)
Invoked to extend spectral-gap analysis from smooth strongly-log-concave to non-smooth targets and to combine with Goyal et al. (2025).

pith-pipeline@v0.9.0 · 5440 in / 1341 out tokens · 26433 ms · 2026-05-16T09:42:31.854420+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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

Proposition 3.2 and Theorems 4.2–4.6: Gap(P) bounded via Cheeger + Υ(δ) F(min π(Si)) from isoperimetric profile of π
IndisputableMonolith/Foundation/ArithmeticFromLogic embed_injective unclear

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

Lemma 4.1, 4.4, 4.5: close-coupling via acceptance probability and Gaussian proposals under non-smooth composite potential

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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.
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