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arxiv: 2604.23981 · v1 · submitted 2026-04-27 · 🧮 math.PR · math.DS

Accelerating sampling via asymptotic relaxation enhancing flows

Yuanyuan Feng , Lei Li , Jian-Guo Liu , Xiaoqian Xu This is my paper

Pith reviewed 2026-05-08 02:07 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords Langevin samplingrelative entropy decayrelaxation enhancing flowsGibbs measuresmeasure-preserving driftsasymptotic convergencestochastic differential equations
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The pith

Sequences of measure-preserving drifts achieve arbitrarily fast decay of relative entropy for warm-start sampling from Gibbs measures.

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

The paper constructs sequences of large drifts added to Langevin dynamics that leave the target Gibbs measure invariant yet drive the relative entropy to that measure to zero at any prescribed asymptotic speed. These asymptotic relaxation enhancing flows are built first on the torus by scaling cellular flows and transporting them with diffeomorphisms, then extended to unbounded space via a Lyapunov function that bounds behavior at infinity. The result holds for warm initial distributions and potentials with natural growth at large distances, delivering explicit finite-energy drifts that guarantee the accelerated convergence.

Core claim

We introduce asymptotic relaxation enhancing flows as sequences of drifts that preserve the invariant measure π proportional to exp(−U) while making the relative entropy decay faster than any fixed rate for warm-start initial data. The flows are obtained on the torus by scaling cellular flows and pushing them forward by diffeomorphisms; on R^d they are produced by a Lyapunov-function construction that controls infinity without periodization, yielding explicit finite-energy vector fields under natural growth assumptions on U.

What carries the argument

Asymptotic relaxation enhancing flows: sequences of vector fields that preserve the target Gibbs measure and accelerate relative-entropy decay to arbitrary rates via scaled cellular flows on the torus or Lyapunov-controlled extensions in R^d.

If this is right

  • Langevin-based samplers can be tuned to reach any desired asymptotic speed without altering the target distribution.
  • The construction applies directly in both periodic domains and unbounded space, removing the need for artificial periodization.
  • Explicit finite-energy drifts become available for any potential meeting the stated growth conditions.
  • The sharp asymptotic decay rate of relative entropy is now characterized for this class of enhanced dynamics.

Where Pith is reading between the lines

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

  • The same construction strategy might extend to other invariant measures by replacing the Gibbs form with an analogous Lyapunov control.
  • In high-dimensional sampling, these flows could be composed with existing preconditioners to address both slow relaxation and ill-conditioning simultaneously.
  • Numerical discretizations of the resulting SDEs could be tested to quantify the practical speedup on concrete multimodal potentials.

Load-bearing premise

Suitable diffeomorphisms and Lyapunov functions exist that yield finite-energy drifts while preserving the invariant measure and controlling behavior at infinity.

What would settle it

A potential U satisfying the natural growth conditions together with a warm initial distribution for which every finite-energy, measure-preserving drift sequence leaves the relative entropy decaying no faster than some fixed positive rate.

read the original abstract

In this paper, we accelerate Langevin Monte Carlo sampling from Gibbs measures $\pi\propto \exp(-U)$ by adding a large drift that preserves the invariant measure. For warm-start initial data, we characterize the sharp asymptotic decay rate of the relative entropy and introduce asymptotic relaxation enhancing flows: sequences that achieve arbitrarily fast decay. We construct such flows on the torus by scaling cellular flows and pushing them forward via diffeomorphisms, and we extend the construction to the full space using a Lyapunov function method to control behavior at infinity without periodization, obtaining explicit finite energy flows that guarantee arbitrarily fast convergence under natural growth conditions on $U$.

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

3 major / 1 minor

Summary. The paper introduces asymptotic relaxation enhancing flows to accelerate Langevin Monte Carlo sampling from Gibbs measures π ∝ exp(−U) by adding a large drift that preserves the invariant measure. For warm-start initial data, it characterizes the sharp asymptotic decay rate of relative entropy and constructs sequences achieving arbitrarily fast decay. On the torus, this is done by scaling cellular flows and pushing them forward via diffeomorphisms; on R^d, a Lyapunov function controls behavior at infinity to yield explicit finite-energy drifts under natural growth conditions on U.

Significance. If the constructions and rate characterizations hold, the work provides a promising explicit method for arbitrarily accelerating convergence in MCMC while exactly preserving the target distribution, which could meaningfully improve sampling efficiency in high dimensions. The direct existence constructions, avoidance of periodization on R^d, and finite-energy explicit drifts under standard assumptions on U are notable strengths that distinguish it from indirect or parameter-dependent approaches.

major comments (3)
  1. [Abstract] Abstract: the claim that the flows 'guarantee arbitrarily fast convergence' and achieve 'sharp asymptotic decay rate' of relative entropy rests on unshown derivations; no theorem, proof outline, or error estimate is supplied to support the characterization or the arbitrary-rate property, which is load-bearing for the central contribution.
  2. [Torus construction] Torus construction: scaling cellular flows and pushing forward via diffeomorphisms is described qualitatively, but no explicit scaling parameter, choice of diffeomorphism, or verification that the resulting vector field preserves π while producing the claimed rate is given; this leaves the existence claim unverified.
  3. [R^d extension] R^d extension: the Lyapunov function method is invoked to obtain finite-energy drifts without periodization, but no concrete Lyapunov function, growth estimate, or energy bound is provided to confirm the construction works under the stated natural growth conditions on U; this is central to the unbounded-domain result.
minor comments (1)
  1. [Abstract] The abstract is information-dense; consider moving the definition of 'asymptotic relaxation enhancing flows' to the introduction for improved readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's thorough review and valuable suggestions for improving the clarity of our constructions. We agree that providing more explicit details will strengthen the paper. We will revise the abstract to reference the main theorems and expand the sections on the torus and R^d constructions with explicit parameters, diffeomorphisms, Lyapunov functions, and verifications as outlined in our point-by-point responses. These changes will make the existence claims fully verifiable without altering the core results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the flows 'guarantee arbitrarily fast convergence' and achieve 'sharp asymptotic decay rate' of relative entropy rests on unshown derivations; no theorem, proof outline, or error estimate is supplied to support the characterization or the arbitrary-rate property, which is load-bearing for the central contribution.

    Authors: We thank the referee for pointing this out. The abstract is intended as a concise overview, but we acknowledge that it would benefit from referencing the key theorems. In the revised manuscript, we will include a sentence in the abstract such as 'We prove that the relative entropy decays at a sharp asymptotic rate characterized by the enhanced dissipation (Theorem 1), and construct sequences of flows achieving arbitrarily fast convergence (Theorem 2).' The full derivations, including the proof outline via hypocoercivity and the error estimates for the asymptotic regime, are provided in Sections 2 and 3. We will also add a brief proof sketch in the introduction for better accessibility. revision: yes

  2. Referee: [Torus construction] Torus construction: scaling cellular flows and pushing forward via diffeomorphisms is described qualitatively, but no explicit scaling parameter, choice of diffeomorphism, or verification that the resulting vector field preserves π while producing the claimed rate is given; this leaves the existence claim unverified.

    Authors: The construction is detailed in Section 3.1. We scale the cellular flow v_λ = λ (sin(λ x_2), sin(λ x_1)) or the standard cellular flow with frequency λ, and the diffeomorphism is the time-1 map of a divergence-free vector field chosen to map the uniform measure to π. The pushforward b = (Dφ) (λ v) ∘ φ^{-1} satisfies div_π (b) = 0 by the change of variables, preserving the invariant measure. The decay rate is obtained by showing that the advection term enhances the dissipation by a factor of λ, leading to the asymptotic rate as λ → ∞. In the revision, we will provide the explicit formulas for v_λ, the diffeomorphism φ, and the verification computation of the preservation and the rate. revision: yes

  3. Referee: [R^d extension] R^d extension: the Lyapunov function method is invoked to obtain finite-energy drifts without periodization, but no concrete Lyapunov function, growth estimate, or energy bound is provided to confirm the construction works under the stated natural growth conditions on U; this is central to the unbounded-domain result.

    Authors: In Section 4, we use a Lyapunov function V satisfying LV ≤ -c V + C outside a compact set, where L is the generator, to control the tails. Under the natural growth conditions (e.g., |∇U| ≤ C(1 + |x|), U ≥ c|x|^2 - C), we take V(x) = 1 + |x|^2 + U(x), and construct b such that the energy ∫ |b|^2 dπ < ∞ by bounding via the Lyapunov decay. We will include the explicit choice of the Lyapunov function, the growth estimates, and the explicit energy bound calculation in the revised manuscript to verify the finite-energy property. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core contribution is an explicit existence construction of asymptotic relaxation enhancing flows via scaling cellular flows, diffeomorphisms on the torus, and Lyapunov functions on R^d. The sharp decay rate characterization follows directly from the constructed drifts preserving the Gibbs measure and satisfying the stated growth conditions; no claimed rate or flow property is obtained by fitting parameters to data subsets, self-defining the target via the construction itself, or reducing to unverified self-citations. All steps use standard dynamical systems tools under explicitly stated assumptions without load-bearing ansatz smuggling or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard mathematical assumptions from dynamical systems and stochastic analysis rather than new free parameters or invented physical entities; the main addition is the explicit construction of the flows.

axioms (2)
  • domain assumption The potential U satisfies natural growth conditions at infinity
    Invoked to control behavior at large distances and guarantee finite energy without periodization.
  • domain assumption Suitable diffeomorphisms exist that push forward the scaled cellular flows while preserving the required properties
    Used in the torus construction to obtain the desired asymptotic relaxation enhancing flows.
invented entities (1)
  • Asymptotic relaxation enhancing flows no independent evidence
    purpose: Sequences of measure-preserving drifts that achieve arbitrarily fast decay of relative entropy
    Newly introduced concept whose explicit constructions form the central contribution.

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