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arxiv: 2605.18069 · v1 · pith:OXHPBJSZnew · submitted 2026-05-18 · 📊 stat.ML · cs.LG· math.PR· math.ST· stat.TH

Wasserstein bounds for denoising diffusion probabilistic models via the F\"ollmer process

Yuta Koike This is my paper

Pith reviewed 2026-05-20 00:33 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.PRmath.STstat.TH
keywords Wasserstein boundsdenoising diffusion probabilistic modelsDDPMFöllmer processsampling error boundsLipschitz scorelog-concave distributionsvariance schedules
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The pith

DDPM sampling error in Wasserstein distance is bounded optimally in dimension and steps by discretizing the Föllmer process under Lipschitz score conditions.

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

The paper establishes sharp upper bounds on the 2-Wasserstein distance between the output of a denoising diffusion probabilistic model and the target distribution. Under general Lipschitz-type conditions on the score function and for broad variance schedules including the cosine schedule, these bounds scale optimally with both data dimension and the number of discretization steps. The analysis replaces the usual reverse Ornstein-Uhlenbeck viewpoint with a discretization of the Föllmer process, which also yields logarithmic Sobolev inequalities and recovers earlier sharp bounds from the literature. For log-concave targets the same optimality holds even without a quadratic transportation cost inequality on the target itself.

Core claim

Viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process, the paper derives sharp 2-Wasserstein error bounds that are optimal in both dimension and number of steps under general Lipschitz-type conditions on the score function. These conditions encompass those commonly imposed on learned scores and imply a logarithmic Sobolev inequality together with a quadratic transportation cost inequality for the DDPM. Consequently, optimal Wasserstein bounds (up to a logarithmic factor) follow from existing sharp KL-divergence bounds under geometric-type variance schedules, and the optimal bound remains attainable for general log-

What carries the argument

Discretization of the Föllmer process, which converts the continuous-time sampling dynamics into a discrete DDPM update while preserving Wasserstein contraction properties under Lipschitz score assumptions.

If this is right

  • Sharp upper bounds on 2-Wasserstein sampling error that are optimal in dimension and number of steps for broad variance schedules including cosine.
  • Recovery of several previously obtained sharp error bounds in the literature.
  • Lipschitz conditions on the score imply a logarithmic Sobolev inequality and quadratic transportation cost inequality for the DDPM.
  • Optimal Wasserstein bound follows from sharp KL-divergence bounds up to a logarithmic factor under geometric schedules.
  • Optimal Wasserstein error remains attainable for general log-concave targets without requiring a quadratic transportation cost inequality on the target.

Where Pith is reading between the lines

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

  • The Föllmer-process viewpoint may extend to analyze convergence of other score-based generative samplers beyond standard DDPM.
  • Cosine and similar schedules may preserve dimension-independent rates more robustly than linear schedules in high-dimensional sampling.
  • Relaxing Lipschitz assumptions while retaining near-optimal rates could be tested on scores learned from image or text data.

Load-bearing premise

The score function satisfies general Lipschitz-type conditions.

What would settle it

A concrete high-dimensional target distribution whose score is Lipschitz yet produces a Wasserstein error that grows with dimension or requires substantially more steps than the derived bound would contradict the optimality claim.

read the original abstract

This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the F\"ollmer process rather than the conventional reverse Ornstein-Uhlenbeck process.

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 establishes sharp 2-Wasserstein error bounds for DDPM sampling by interpreting the sampler as a discretization of the Föllmer process. Under general Lipschitz-type conditions on the score, it derives dimension- and step-optimal upper bounds for a broad class of variance schedules (including cosine), recovers prior sharp results, shows that the same conditions imply LSI and quadratic transportation-cost inequalities, and extends the optimal Wasserstein bound to general log-concave targets even without a T2 inequality for the target.

Significance. If the central derivations hold, the work supplies a unified, process-level framework that yields optimal Wasserstein rates without hidden dimension dependence for standard schedules and recovers existing sharp bounds as special cases. The Föllmer-process perspective and the explicit control of discretization error via dimension-uniform Gronwall estimates are technically valuable contributions to the analysis of diffusion samplers.

major comments (2)
  1. [§3.2, Theorem 3.1] §3.2, Theorem 3.1: the claimed dimension-free constant in the discretization error bound for the cosine schedule relies on a specific Gronwall-type estimate; the proof sketch should explicitly track the dependence on the Lipschitz constant L and the schedule parameter to confirm uniformity in d.
  2. [§4.1, Eq. (12)] §4.1, Eq. (12): the passage from the Lipschitz score assumption to the LSI constant appears to use a standard Bakry-Émery argument, but the resulting transportation-cost inequality constant should be stated explicitly so that the subsequent Wasserstein bound can be compared directly with the KL-based bound of prior work.
minor comments (2)
  1. [Abstract] The abstract states that the bounds are 'optimal in both the dimension and the number of steps' but does not indicate the precise dependence on the number of steps N; a short parenthetical remark would improve readability.
  2. [§2 and §5] Notation for the variance schedule (β_t versus the cosine form) is introduced in §2 but reused without redefinition in §5; a single consolidated definition would prevent minor confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.1] the claimed dimension-free constant in the discretization error bound for the cosine schedule relies on a specific Gronwall-type estimate; the proof sketch should explicitly track the dependence on the Lipschitz constant L and the schedule parameter to confirm uniformity in d.

    Authors: We agree that an explicit tracking of the dependence on the Lipschitz constant L and the schedule parameter in the Gronwall estimate will strengthen the presentation and confirm the claimed dimension-free property. In the revised version we will expand the proof sketch of Theorem 3.1 in §3.2 to display these dependencies step by step, showing that the resulting constant remains independent of dimension d for the cosine schedule (and more generally for the class of schedules considered). revision: yes

  2. Referee: [§4.1, Eq. (12)] the passage from the Lipschitz score assumption to the LSI constant appears to use a standard Bakry-Émery argument, but the resulting transportation-cost inequality constant should be stated explicitly so that the subsequent Wasserstein bound can be compared directly with the KL-based bound of prior work.

    Authors: We thank the referee for this observation. The derivation in §4.1 indeed applies the Bakry-Émery criterion to the Lipschitz score assumption to obtain the logarithmic Sobolev inequality, from which the quadratic transportation-cost inequality follows with an explicit constant that depends on the Lipschitz constant and the variance schedule. In the revised manuscript we will state this constant explicitly after Eq. (12), enabling a direct comparison with the constants appearing in the KL-based Wasserstein bounds of prior work. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation views DDPM sampling as discretization of the Föllmer process under external Lipschitz-type score conditions that imply LSI and quadratic transportation inequalities via standard implications. Discretization error is bounded by Gronwall-type estimates that are uniform in dimension for the given variance schedules (including cosine). Optimality claims recover prior literature results without reducing any prediction or bound to a fitted parameter or self-citation chain; the central Wasserstein bounds follow from the stated assumptions and external stochastic-process tools rather than internal redefinition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on Lipschitz-type conditions on the score and on properties of the Föllmer process; these are standard domain assumptions rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Score function satisfies general Lipschitz-type conditions
    Invoked to obtain sharp bounds and to imply logarithmic Sobolev inequality; stated in contribution (i) and (ii).

pith-pipeline@v0.9.0 · 5748 in / 1243 out tokens · 42341 ms · 2026-05-20T00:33:33.786206+00:00 · methodology

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