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Counterexample / Generative models

Incorrect Gaussian noise scaling in the DPS Jensen-gap bound

Diffusion Posterior Sampling for General Noisy Inverse Problems ↗

1,867 citations ↗Semantic Scholar · 2026-09-09

2023 · arXiv:2209.14687v4, 20 May 2024; ICLR 2023 · Reviewed 09 September 2026

An explicit example contradicts the selected statement as written.

Paper context

Overview

Diffusion Posterior Sampling uses a pretrained diffusion model to reconstruct signals or images from noisy measurements, including nonlinear measurement processes. It combines diffusion sampling with measurement guidance computed through a denoised estimate of the unknown clean input. A key approximation evaluates the measurement likelihood at that estimate instead of averaging it over the entire conditional distribution.

Role of the theoretical result

Theorem 1 is intended to quantify the likelihood-level error introduced by the posterior-mean approximation under Gaussian measurement noise. Its bound is part of the justification for that approximation, but it does not directly bound the guidance-gradient error used by the sampler.

Original paper ↗

01 / Summary

Summary of the result

The Gaussian likelihood constant in Lemma 2, propagated into Theorem 1, has incorrect dependence on the noise standard deviation. For identity observation, σ=0.1, y=0.2 and conditional state distribution N(0,0.01), the signed Jensen gap is approximately 0.497859. Equation (14) gives an upper bound of approximately 6.14×10⁻²³. The univariate likelihood has Lipschitz constant exp(−1/2)/(√(2π)σ²); substituting this constant repairs the density-bound argument.

02 / Statement

Statement under review

Definition 1, Eq. (12), PDF p. 4; Theorem 1, Eq. (14), p. 5; Lemmas 2–3 and proof of Theorem 1, pp. 14–15. · paraphrased

Theorem 1 bounds the Jensen gap between the conditional likelihood and its posterior-mean approximation by a Gaussian-noise-dependent constant times the forward operator's derivative bound and the posterior's first centered absolute moment.

J=E[p(y∣X0)∣Xt]−p(y∣E[X0∣Xt]),J≤d2πσ2e−1/(2σ2)∥∇A∥m1.\mathcal J=\mathbb E[p(y\mid X_0)\mid X_t]-p(y\mid\mathbb E[X_0\mid X_t]),\qquad \mathcal J\le\frac{d}{\sqrt{2\pi\sigma^2}}e^{-1/(2\sigma^2)}\|\nabla\mathcal A\|m_1.
Relevant assumptions ↘
  • The measurement noise is Gaussian with standard deviation σ>0.
  • The forward operator has a finite derivative bound.
  • The conditional distribution of X0 given Xt has a finite first centered absolute moment m1.
  • The witness is one-dimensional and comes from an ordinary variance-preserving Gaussian forward diffusion with an exact conditional mean.
  • The Jensen gap is signed as defined in Eq. (12); the witness below makes it positive.

03 / Derivation

Counterexample and derivation

6 steps · complete derivation
  1. 01

    Use a realizable Gaussian conditional distribution

    Take independent standard normals X0 and Z and Xt=√0.99 X0+√0.01 Z. At Xt=0, Gaussian conditioning gives X0|Xt=0 distributed as N(0,0.01), with conditional mean zero.

    X0,Z∼indN(0,1),Xt=0.99X0+0.01Z,X0∣Xt=0∼N(0,0.01).X_0,Z\stackrel{\mathrm{ind}}{\sim}\mathcal N(0,1),\quad X_t=\sqrt{0.99}X_0+\sqrt{0.01}Z,\quad X_0\mid X_t=0\sim\mathcal N(0,0.01).
  2. 02

    Set the measurement model

    Use the identity forward operator, measurement standard deviation 0.1, and observed value y=0.2. The operator bound is 1 and the centered first moment is explicit.

    A(x)=x,σ=0.1,y=0.2,m1=0.12/π.\mathcal A(x)=x,\quad \sigma=0.1,\quad y=0.2,\quad m_1=0.1\sqrt{2/\pi}.
  3. 03

    Integrate the likelihood exactly

    Convolving the conditional N(0,0.01) distribution with N(0,0.01) measurement noise produces an N(0,0.02) likelihood for y. The plug-in approximation instead evaluates the N(0,0.01) density at y.

    p(0.2∣Xt=0)=e−10.04π,p(0.2∣X^0=0)=e−20.02π.p(0.2\mid X_t=0)=\frac{e^{-1}}{\sqrt{0.04\pi}},\qquad p(0.2\mid\widehat X_0=0)=\frac{e^{-2}}{\sqrt{0.02\pi}}.
  4. 04

    Compare the signed gap with the printed bound

    The gap is positive, approximately 0.497859. The theorem's right-hand side is approximately 6.14×10⁻²³.

    J=e−10.04π−e−20.02π≈0.4978590784>e−50π≈6.139401446×10−23.\mathcal J=\frac{e^{-1}}{\sqrt{0.04\pi}}-\frac{e^{-2}}{\sqrt{0.02\pi}}\approx0.4978590784>\frac{e^{-50}}{\pi}\approx6.139401446\times10^{-23}.
  5. 05

    Differentiate the Gaussian density

    For φσ(r)=(√(2π)σ)⁻¹exp(−r²/(2σ²)), the magnitude of the derivative is maximized at |r|=σ. This gives the correct one-dimensional Lipschitz constant.

    ∣ϕσ′(r)∣=∣r∣2πσ3e−r2/(2σ2),sup⁡r∣ϕσ′(r)∣=e−1/22πσ2.|\phi_\sigma'(r)|=\frac{|r|}{\sqrt{2\pi}\sigma^3}e^{-r^2/(2\sigma^2)},\qquad \sup_r|\phi_\sigma'(r)|=\frac{e^{-1/2}}{\sqrt{2\pi}\sigma^2}.
  6. 06

    Recover a valid first-moment estimate

    Apply the corrected Lipschitz bound to the likelihood composed with the forward operator, then average the distance from the conditional mean. This bounds the absolute gap and therefore also its signed value.

    ∣J∣≤e−1/22πσ2∥∇A∥m1.|\mathcal J|\le\frac{e^{-1/2}}{\sqrt{2\pi}\sigma^2}\|\nabla\mathcal A\|m_1.

Counterexample

Under the stated Gaussian construction, E[p(y|X₀)]−p(y|E[X₀])=0.49785908>6.14×10⁻²³. The gap uses the signed convention in Definition 1; its positivity is essential to this counterexample.

σ=0.1, y=0.2, X0∣Xt=0∼N(0,0.01):J≈0.497859>6.1394×10−23.\sigma=0.1,\ y=0.2,\ X_0\mid X_t=0\sim\mathcal N(0,0.01):\qquad \mathcal J\approx0.497859>6.1394\times10^{-23}.
Numerical illustration

Noise dependence of the signed Jensen gap

0.00.51.0
Exact signed Jensen gap0.4979
Printed upper bound6.14e-23

Set y = 2σ and X₀ | Xₜ = 0 ∼ N(0, σ²), with identity measurement. The gap is E[p(y | X₀)] − p(y | E[X₀]), exactly the paper’s signed convention. Every setting shown violates the printed upper bound. This tests a density bound, not reconstruction quality.

Solid: exact gap. Dashed: printed bound. Horizontal axis: σ from 0.05 to 0.5.

04 / Implications

Implications and proposed correction

Theoretical implications

Affected result

The printed Eq. (14) fails even for a one-dimensional Gaussian forward diffusion and a linear observation operator. The univariate Gaussian constant in Lemma 2 is also incorrect.

Empirical scope

Relation to reported experiments

This is not an empirical test of DPS reconstructions. It does not show that the sampler fails on the reported tasks. The separate limitation that a density approximation does not by itself control the score has already been discussed in later work and is not presented here as an unresolved finding.

Proposed correction

Sufficient conditions and revised bound

Replace the univariate likelihood Lipschitz constant by e^(−1/2)/(√(2π)σ²). In n measurement dimensions with Euclidean norms, the isotropic Gaussian density has gradient supremum e^(−1/2)/((2π)^(n/2)σ^(n+1)); combine it with an explicit operator Lipschitz constant and m1.

∣J∣≤e−1/2(2π)n/2σn+1Lip⁡(A) m1.|\mathcal J|\le\frac{e^{-1/2}}{(2\pi)^{n/2}\sigma^{n+1}}\operatorname{Lip}(\mathcal A)\,m_1.

Implementation implications

A certificate or error tolerance based on the printed noise constant needs correction. The mathematical repair does not change the DPS denoiser or guidance update. A guarantee on the score, which the algorithm uses, still requires a separate argument.

Limits of this review

  • Likelihood densities can exceed one; the displayed likelihood values are densities, not event probabilities.
  • The positive signed-gap witness uses y=0.2. At y=0, the corresponding signed gap is negative and would not by itself refute Eq. (14).
  • The conditional Gaussian is the standard regular conditional distribution, and the violation persists for a neighborhood of the displayed observation and diffusion state.

05 / References

Sources and correction history

  1. 01
    DPS, exact version ↗

    Definition 1/Eq. (12), p. 4; Theorem 1/Eq. (14), p. 5; Lemmas 2–3 and theorem proof, pp. 14–15

  2. 02
    Version history ↗

    Latest listed revision v4, 20 May 2024

  3. 03
    Author repository discussions ↗

    Available open and closed issues and comments checked during the audit

  4. 04
    Frequency-Guided Posterior Sampling for Diffusion-Based Image Restoration ↗

    Gaussian conditional/score analysis, Section 5 and Appendix B

  5. 05
    On the Guidance of Flow Matching ↗

    Section 3.2, discussion following the local guidance error bound; density error versus guidance-vector-field error

  6. 06
    ICLR discussion page ↗

    Discussion access was incomplete in this audit

Download the arithmetic witnesses · Python, no dependencies ↓
Correction search · 09 September 2026

2026-09-09 The signed-gap definition, theorem, Gaussian lemmas, and proof were reread; the theorem page was visually checked. Current arXiv history and targeted searches for the Gaussian Lipschitz constant and theorem correction were checked. Available author repository issues and comments and the indicated later Gaussian/score analyses were inspected during the audit. "Diffusion Posterior Sampling" "Lipschitz constant" error "Diffusion Posterior Sampling" "Lemma 2" correction "Diffusion Posterior Sampling" "Gaussian" "bound" "incorrect" No source correcting this exact Gaussian constant was found in the bounded search. Later work discusses Gaussian conditional covariance and the inability to derive a score bound directly from a density bound. Those concerns are treated as related, already discussed limitations and are excluded from this finding. The original OpenReview discussion and notes endpoints were not fully accessible. An indexed later document reproduced the disputed constant, but its complete identity and text were not verified; no downstream-propagation claim is made from that result. The literature search is bounded and does not establish novelty.

A bounded search is not evidence of priority or proof that no correction exists.

Entirely AI-generated analysis, including cross-checks by separate AI agents; no independent human verification. Authors have not been contacted. Review standard.

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