Sampling Binary Data by Denoising through Score Functions

Francis Bach; Saeed Saremi

REVIEW 4 major objections 7 minor 35 references

Sampling Binary Data by Denoising through Score Functions

T0 review · 4 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read On the hypercube, the optimal denoiser is the sign of the score, and discrete Langevin sampling converges exponentially at high noise.

desk verdict A careful theory paper on binary walk-jump sampling whose contraction bounds are new and sound, but whose end-to-end claim is unproven because the learned score never appears in the analysis. read the letter →

arxiv 2502.00557 v1 pith:47XN5W5F submitted 2025-02-01 stat.ML cs.LG

Francis Bach , Saeed Saremi This is my paper · ORCID

classification stat.MLcs.LG MSC 60J2260J05

keywords binarydataBernoullinoiseTweedie-MiyasawaformuladenoisingdiscreteLangevinsamplingBooleanhypercubemulti-measurementWassersteinconvergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish that binary data on the hypercube $\{-1,1\}^d$ can be learned and sampled without leaving the discrete world. The key move is to smooth the target distribution $p$ by independent Bernoulli sign flips, producing a noisy law $q_\alpha$, and to prove the identity $\mathbb{E}[x|y] = (1/\alpha)\nabla\log q_\alpha(y)$ for $y = x \circ \varepsilon$. That identity makes the optimal Hamming-loss denoiser equal to the sign of the score, exactly as the Tweedie-Miyasawa formula does for Gaussian noise, and it turns denoising into score estimation. The paper then analyzes discrete Langevin samplers on $q_\alpha$, showing exponential Wasserstein contraction at high noise, and a multi-measurement extension that lowers effective noise by averaging several noisy observations at a fixed level. If the claims hold, binary generative models could be trained by logistic-regression denoising and sampled by a fixed-noise Markov chain, with no diffusion time discretization.

What carries the argument

The carrying object is the binary Tweedie-Miyasawa identity $\mathbb{E}[x|y]=(1/\alpha)\nabla\log q_\alpha(y)$ for the Bernoulli sign-flip noise model. The score $\nabla\log q_\alpha$ is a continuous function on $\mathbb{R}^d$ obtained from an exponential tilt of $p$, and its regularity is controlled by $\|\nabla\log q_\alpha\|_\infty\le \alpha$ and the $\ell^1$-Lipschitz bound $\|\nabla\log q_\alpha(y)-\nabla\log q_\alpha(y')\|_\infty \le \alpha^2\|y-y'\|_1$, which come from $\nabla^2\log q_\alpha=\alpha^2\operatorname{cov}(x|y)$. These constants turn the sampling problem into a contraction problem for discrete Langevin kernels, and the multi-measurement variant uses the averaged argument $\bar y_{1:m}$ to reduce effective noise.

What would settle it

For the $d=8$ mixture model used in the paper, compute the exact transition matrix of the two-stage sampler, run it from uniform initialization, and compare the empirical Wasserstein distance to $q_\alpha$ with the bound $W(q',q)\le 12d\sqrt{\beta_2 d}$ and the contraction rate $1-\tfrac12 e^{-2/\eta-2\beta_1}$ under the stated conditions; a violation at allowed parameters would falsify the quantitative claim. A complementary check is to evaluate a trained denoiser at random averaged inputs $\bar y_{1:m}\in[-1,1]^d$ and compare with exact $\mathbb{E}[x|y_{1:m}]$ for a synthetic prior, testing whether the learned score is accurate where multi-measurement sampling uses it.

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Extended reading notes

Core claim

For the random sign-flip model $y = x \circ \varepsilon$ with $\Pr(\varepsilon_i=1)=\sigma(2\alpha)$, the paper's central identity is $\mathbb{E}[x|y] = (1/\alpha)\nabla\log q_\alpha(y)$, where $q_\alpha(y) \propto \sum_{x} p(x) e^{\alpha x^\top y}$ is the law of the noisy data, defined for all $y\in\mathbb{R}^d$ even though the data live on the hypercube. Hence the optimal denoiser under Hamming loss is $f(y)=\operatorname{sign}(\mathbb{E}[x|y])=\operatorname{sign}(\nabla\log q_\alpha(y))$, and the score can be learned by logistic regression on pairs $(x, x\circ\varepsilon)$. For sampling, the paper introduces one-stage and two-stage discrete Langevin kernels and proves exponential contraction in the Wasserstein metric when $4\beta_2 d e^{2\beta_1}\le 1$ (respectively $8\beta_2 d e^{4\beta_1}\le 1$), with $\beta_1=\alpha$, $\beta_2=\alpha^2$ for the true score; the two-stage sampler has stationary bias at most $12d\sqrt{\beta_2 d}$ when $e^{-2/\eta+2\beta_1}\le 1/d$. For $m$ independent noisy measurements, $\mathbb{E}[x|y_{1:m}]=(1/m\alpha)\nabla\log q_{m\alpha}(\bar y_{1:m})$, so the effective noise is $m\alpha$ while each sampling step stays at level $\alpha$.

Load-bearing premise

The load-bearing premise is that the denoiser used in practice is close enough to the true mathematical score, with the same boundedness and smoothness, even at points inside the hypercube where it is only trained indirectly; the proofs assume that exact score, while the experiments run with a learned approximation.

Editorial extensions

If this is right

Training a binary generative model reduces to logistic regression on noisy hypercube pairs, with no continuous-time diffusion or noise schedule, provided the learned denoiser approximates $\nabla\log q_\alpha$.
At noise level $\alpha\le 1/(4\sqrt d)$, the one-stage discrete Langevin chain contracts exponentially in Wasserstein distance, so its mixing time is controlled by the rate $1-\tfrac12 e^{-2/\eta-\alpha}$.
With $m$ independent noisy measurements, the denoising error bound improves from $d e^{-2\alpha}$ to $d e^{-m\alpha}$, while the sampler can still run at the easier fixed noise level $\alpha$.
The two-stage sampler has stationary bias at most $12d\sqrt{\beta_2 d}=12\alpha d^{3/2}$ when $e^{-2/\eta+2\alpha}\le 1/d$, which is small compared with the hypercube diameter once $\alpha\ll 1/\sqrt d$; this is the paper's formal analogue of log-concave sampling in Euclidean space.
On binarized MNIST, step size $\eta=1/\alpha$ gives fast mixing at $\alpha=0.5$ and breakdown at $\alpha=2$, consistent with the high-noise regime the theory covers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the same exponential-tilt structure should extend to other exponential-family noise channels, where the effective noise parameter would play the role of $\alpha$; the contraction constants would need to be re-derived for each channel.
Extension: a practical test of the theory is to estimate the Lipschitz constant of trained denoisers at interior points $\bar y_{1:m}\in[-1,1]^d$; if it substantially exceeds $\alpha^2$, the guaranteed contraction regime may shrink and the two-stage advantage could vanish.
Extension: because multi-measurement sampling evaluates the score at averaged points inside the hypercube, training data drawn only from vertices may under-cover that region; generating corrupted-and-averaged inputs during training is a direct way to close the gap.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new material is the Wasserstein contraction analysis for discrete Langevin samplers (Props 3.1 and 3.3) and the clean binary Tweedie-Miyasawa identity (Lemma 2.2). The multi-measurement extension is a direct analog of the Gaussian work you know, but the contraction bounds are a real addition to the discrete sampling literature. The appendices are self-contained and careful; the regularity assumptions are stated and verified for the exact q_α. So the idealized framework is sound.

The soft spot is exactly where the stress-test note lands: every convergence result is about a chain driven by the exact score s = ∇ log q_α. The method replaces s with a U-Net trained on hypercube vertices, and the paper never bounds ∥s_learned − s∥ or shows that the learned score inherits the ℓ∞ bound α and the α²-Lipschitz condition. If those constants are exceeded, the contraction proofs in Props 3.1 and 3.3 do not apply. For a U-Net there is no reason to expect them to hold. That is a real gap between the title's promise and what is proven. I'd call it a gap, not a fatal flaw: the idealized analysis is honest and a perturbation theorem would be a natural follow-up.

The second soft spot is the regime mismatch. The theory needs α ≲ 1/√d or 1/d for contraction and stationarity, which is very high noise; denoising at that noise is poor, with Lemma 2.3 giving only de^{-2α}. The MNIST experiments at α = 0.5 (d = 784) sit outside the stated conditions, and η = 1/α violates the Prop 3.4 condition e^{-2/η+2β1} ≤ 1/d. Those experiments are qualitative with no baselines, so they carry little evidential weight. The synthetic experiments are more informative but mostly at small d.

This is an honest theory contribution, not a demonstration of a working generative model. It deserves a serious referee: the contraction bounds are new and the proof strategy is reusable. The referee should ask for a score-error propagation section, or at least an explicit statement that the end-to-end claim is not yet theorem-backed. The citation practice is fine; the self-citations are to the Gaussian counterpart the work directly extends. If I were the editor, I'd send it to review, expecting heavy revision on the experimental claims.

Referee Report

4 major / 7 minor

Summary. The paper studies Bernoulli sign-flip smoothing y = x ∘ ε on {−1,1}^d, defines an extension q_α(y) of the noisy pmf to R^d via q_α(y) ∝ Σ_x p(x) e^{α x^⊤ y}, proves a Tweedie–Miyasawa identity E[x|y] = (1/α)∇ log q_α(y) (Lemma 2.2), proposes logistic-regression denoising to learn the score, and introduces one- and two-stage discrete Langevin samplers. It proves Wasserstein contraction and stationary-bias bounds for the exact-score chains (Propositions 3.1–3.4), extends the identity to multiple measurements with effective noise mα (Lemma 2.4), and reports synthetic and binarized-MNIST experiments.

Significance. If the theoretical claims held for the implemented algorithm, this would be a valuable alternative to discrete diffusion: it avoids SDE discretization, gives an exact denoising–score identity on the hypercube, and provides explicit Wasserstein contraction results for discrete Langevin-like kernels in this setting. The appendices are careful and the lemmas and propositions are proved from stated assumptions with explicit constants. The main caveat is that the theory is oracle-level: it uses the exact score, whereas the experiments use a learned U-Net, and the demonstrated operating points lie outside the theorem regimes. The contribution is nonetheless a solid conceptual and algorithmic framework.

major comments (4)

[Section 3, Propositions 3.1–3.4] All four convergence results are proved for the exact score s = ∇ log q_α satisfying Eq. (4) and Eq. (8). The implemented method replaces s by a U-Net trained by logistic regression on hypercube vertices. No bound on ∥ŝ − s∥ is given, and it is not shown that ŝ inherits ∥ŝ∥∞ ≤ α or an α²-Lipschitz property. Since the stationary distribution in Propositions 3.2 and 3.4 is defined through a transition kernel built from the exact score, the end-to-end claim of "sampling binary data by denoising through score functions" is proved only for an oracle, not for the algorithm in Section 5.
[Sections 2.4 and 3.2] Multi-measurement and sequential sampling require the score at interior points ȳ ∈ [−1,1]^d and at parameter mα. Section 2.3's training procedure uses single noisy vertices only. The remark in Section 2.4 that one can train on averaged measurements is not supported by experiments or analysis, and Section 3 correctly notes that the extension of q_α to R^d is non-unique up to additive linear score terms. Thus the learned denoiser's behavior on the interior, on which Lemma 2.4 and the sequential sampler rely, is unconstrained and unverified.
[Section 5.2 and Propositions 3.3–3.4] The MNIST experiments are outside the stated theorem conditions. For d = 784 and α = 0.5, the condition 8dβ₂e^{4β₁} ≤ 1 of Propositions 3.3 and 3.4 gives 8 · 784 · 0.25 · e² ≈ 1.16 × 10⁴, and η = 1/α gives e^{−2/η + 2β₁} = 1, violating the assumption e^{−2/η + 2β₁} ≤ 1/d of Proposition 3.4. The claim that the experiments validate the theoretical findings is therefore unsupported; at best they are heuristics in a regime the theory does not cover.
[Lemma 2.3 versus Propositions 3.1–3.4] There is a mismatch between the regime where the sampler is proven to converge and the regime where denoising is useful. Lemma 2.3 gives denoising error O(d e^{−2α}), which is small only for large α, while the contraction conditions hold only for α ≪ 1/d (one-stage) or α ≪ 1/√d (two-stage). No composite end-to-end bound quantifies the trade-off for the procedure "sample from q_α, then denoise"; the multi-measurement idea in Lemma 2.4 is the right remedy, but its combination with the sequential sampler's bias is not analyzed.

minor comments (7)

[Section 2.1, Eq. (1)] The index set in "for i ∈ {-1, . . . ,1}^d" should be i ∈ {1, . . . , d}.
[Appendix A] The proof of Lemma 2.4 says "Following the same reasoning that in the proof of Lemma 2.4"; the intended reference is Lemma 2.3.
[Section 3, Eq. (4)] In the Lipschitz condition, "∥y − y∥₁" should read "∥y − y′∥₁".
[Section 5.1 and Section 5.2] The text uses "learning rate equal to 1/α" in Section 5.1 and "step-size η" elsewhere; the terminology should be unified.
[Figure 3 and surrounding text] The mixing time is reported as 1/(1 − λ₂), but λ₂ is not defined in the main text; it should be defined explicitly.
[Appendices B–E] The use of Wikipedia links for the total variation distance, Chernoff bound, and Pinsker's inequality is informal; standard textbook references (e.g., Levin and Peres, 2017) would be more appropriate.
[Section 5.2] The qualitative MNIST panels would benefit from quantitative metrics, especially because the theoretical bounds do not apply in that regime.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central identities and convergence proofs are self-contained; self-citations serve only as template and comparison, not as load-bearing evidence.

full rationale

The paper's central identities (Lemma 2.2 and Lemma 2.4) are algebraic consequences of the exponential-tilt definition of q_alpha, not definitions of the target in terms of the method: the conditional expectation E[x|y] and the score function grad log q_alpha are independently defined, and the equality is proved from Eq. (3). The convergence results (Propositions 3.1-3.4) are proved from explicit regularity assumptions (Eqs. (4) and (8)) that are verified for the true q_alpha; no fitted parameter is renamed as a prediction. The frequent citations to the authors' prior work supply the Gaussian walk-jump template and provide comparisons, but the binary extension is derived in this paper rather than imported; no load-bearing claim rests on an unverified self-citation. The gap between the exact-score theory and the learned-score experiments is a correctness risk about score estimation error and regularity of the learned score, not a circularity in the derivation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The framework introduces no new physical or mathematical entities; it reuses the score function and the discrete Langevin kernel from prior work. The free parameters are the noise level α and step-size η, both user-chosen hyperparameters. The axioms are the regularity conditions needed for the convergence proofs and the binary hypercube setting.

free parameters (2)

noise level α = user-specified (e.g., 0.5 for MNIST)
Controls the Bernoulli noise strength σ(2α); all theoretical bounds scale with α, and the recommended step-size is η = 1/α. Not fitted to data but is the key hyperparameter.
step-size η = set to 1/α in experiments
Chosen by hand to balance mixing speed and bias; the theory in Prop. 3.4 requires a smaller η for the bias bound to be meaningful.

assumptions (4)

domain assumption The score s satisfies the regularity conditions Eq. (4): ||s(y)||∞ ≤ β1 and ||s(y) - s(y')||∞ ≤ β2||y - y'||1.
The paper shows β1 = α, β2 = α² for the true q_α, but the learned score is only assumed to satisfy these; violations would void Props. 3.1-3.4.
domain assumption Smooth convexity Eq. (8): 0 ≤ log q(y) - log q(z) - s(z)ᵀ(y - z) ≤ (β2/2)||y - z||₁².
Used in Prop. 3.4 to bound the bias of the two-stage sampler; stated to hold for q_α.
standard math The noisy distribution q_α, defined by the exponential tilt in Eq. (2), is a valid density extension to R^d so that gradients can be taken on the interior of the hypercube.
Needed for the score ∇ log q_α at non-vertex points in multi-measurement sampling; it is a mathematical construction, not an empirical fact.
domain assumption Data lies on the Boolean hypercube {−1,1}^d and the noise is independent Bernoulli sign flips.
The entire framework is restricted to binary data; the paper does not address continuous or categorical data.

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Cite this review

Pith. "Pith review of Sampling Binary Data by Denoising through Score Functions." pith.science (2026). https://pith.science/paper/47XN5W5F

@misc{pith2026250200557,
  author       = {Pith},
  title        = {Pith review of: Sampling Binary Data by Denoising through Score Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47XN5W5F}},
  note         = {Machine review of arXiv:2502.00557}
}

read the original abstract

Gaussian smoothing combined with a probabilistic framework for denoising via the empirical Bayes formalism, i.e., the Tweedie-Miyasawa formula (TMF), are the two key ingredients in the success of score-based generative models in Euclidean spaces. Smoothing holds the key for easing the problem of learning and sampling in high dimensions, denoising is needed for recovering the original signal, and TMF ties these together via the score function of noisy data. In this work, we extend this paradigm to the problem of learning and sampling the distribution of binary data on the Boolean hypercube by adopting Bernoulli noise, instead of Gaussian noise, as a smoothing device. We first derive a TMF-like expression for the optimal denoiser for the Hamming loss, where a score function naturally appears. Sampling noisy binary data is then achieved using a Langevin-like sampler which we theoretically analyze for different noise levels. At high Bernoulli noise levels sampling becomes easy, akin to log-concave sampling in Euclidean spaces. In addition, we extend the sequential multi-measurement sampling of Saremi et al. (2024) to the binary setting where we can bring the "effective noise" down by sampling multiple noisy measurements at a fixed noise level, without the need for continuous-time stochastic processes. We validate our formalism and theoretical findings by experiments on synthetic data and binarized images.

Figures

Figures reproduced from arXiv: 2502.00557 by the authors.

**Figure 2.** Optimal denoising from multiple measurements, for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Comparison of 1-stage and 2-stage Langevin sampling. Top: distance to desired distribution [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Comparison of 1-stage and 2-stage Langevin sampling. Top: distance to desired distribution [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: The denoising performance on binarized MNIST at two high Bernoulli noise levels ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: The sampling performance of our algorithm for binarized MNIST at three Bernoulli noise levels, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Austin, J., Johnson, D., Ho, J., Tarlow, D., and van den Berg, R. (2021). Structured denoising diffusion models in discrete state-spaces. In Advances in Neural Information Processing Systems

work page 2021

[2] [2]

Bach, F. (2024). Learning Theory from First Principles . MIT Press

2024

[3] [3]

Campbell, A., Benton, J., De Bortoli, V., Rainforth, T., Deligiannidis, G., and Doucet, A. (2022). A continuous time framework for discrete denoising models. In Advances in Neural Information Processing Systems

work page 2022

[4] [4]

S., Bartlett, P

Cheng, X., Chatterji, N. S., Bartlett, P. L., and Jordan, M. I. (2018). Underdamped L angevin MCMC : A non-asymptotic analysis. In Conference on Learning Theory

work page 2018

[5] [5]

Chewi, S. (2024). Log-concave Sampling . Book draft

work page 2024

[6] [6]

Dalalyan, A. S. (2017). Theoretical guarantees for approximate sampling from smooth and log-concave densities. Journal of the Royal Statistical Society. Series B (Statistical Methodology) , 79(3):651--676

work page 2017

[7] [7]

and Nichol, A

Dhariwal, P. and Nichol, A. (2021). Diffusion models beat GAN s on image synthesis. In Advances in Neural Information Processing Systems

work page 2021

[8] [8]

and Moulines, \'E

Durmus, A. and Moulines, \'E . (2017). Nonasymptotic convergence analysis for the unadjusted Langevin algorithm . The Annals of Applied Probability , 27(3):1551 -- 1587

work page 2017