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arxiv: 2606.08309 · v1 · pith:ZHQNLHHUnew · submitted 2026-06-06 · 💻 cs.LG · cs.CV

Where the Score Lives: A Wavelet View of Diffusion

Emma Finn , Binxu Wang , T. Anderson Keller , Demba E. Ba This is my paper

Pith reviewed 2026-06-27 19:59 UTC · model grok-4.3

classification 💻 cs.LG cs.CV
keywords score-based generative modelsdiffusion modelswavelet basisscore functiondata momentsdenoisinginductive biasesgenerative behavior
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The pith

Expanding the score function in a 2D wavelet basis makes it analytically solvable in terms of the moments of the data distribution.

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

The paper proposes a parameterization of the score function for diffusion models using an expansion in a 2D orthogonal wavelet basis. This yields optimal score functions that are explicit functions of the moments of the data. A sympathetic reader would care because it offers an architecture-independent way to analyze which features of the data distribution are most important for effective denoising. It also shows how this approach can reproduce some behaviors of common networks like U-Nets without committing to their specific design. This helps explain why different score approximators produce different generative results.

Core claim

By expanding the score in a 2D orthogonal wavelet basis, the authors obtain an analytically solvable parameterization whose coefficients are directly determined by the moments of the data distribution. This moment-based form is interpretable and flexible enough to partially mimic the inductive biases of architectures such as U-Nets and CNNs, providing an architecture-agnostic view of what attributes matter most for denoising in score-based generative models.

What carries the argument

The 2D orthogonal wavelet basis expansion of the score function, which turns it into an explicit sum over moment-derived coefficients.

If this is right

  • The moment coefficients reveal which attributes of the data distribution matter most for denoising.
  • This parameterization can reproduce relevant inductive biases of U-Nets and CNNs.
  • Researchers can analyze how the data distribution interacts with the score network independently of architecture choice.
  • Distinct generative behaviors across architectures can be traced to differences in how they approximate these moment-based scores.

Where Pith is reading between the lines

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

  • This approach might allow construction of new score networks that directly incorporate moment calculations for improved interpretability.
  • Testing the wavelet scores on datasets with known moment structures could show whether they match or exceed standard diffusion performance.
  • Connections to other basis expansions like Fourier might reveal similar moment-based insights in different domains.

Load-bearing premise

That expanding the score in wavelets produces coefficients based on moments that capture the data attributes essential for good denoising performance.

What would settle it

Running the wavelet-based score on standard image datasets and finding that the generated samples have significantly worse quality metrics than those from a U-Net score network would indicate the parameterization does not capture the necessary attributes.

Figures

Figures reproduced from arXiv: 2606.08309 by Binxu Wang, Demba E. Ba, Emma Finn, T. Anderson Keller.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Wavelet Expansion of the Score. The score of a given image Xt can be expanded in an or￾thonormal wavelet basis by representing it as a sum of coefficients ci(Xt) multiplied by corresponding wavelets wi (Equation 5). Since the wavelets come in many translated copies, we only display wavelets at a single spatial location k, and display coefficients in spatially ordered ‘feature maps’. The sum over spatial lo… view at source ↗
Figure 3
Figure 3. Figure 3: Wavelet Coefficient Approximation. We approximate the unknown true coefficients ci(Xt) of the true score function as an inner product of features φi(Xt) and parameters α (t) i . The features depicted here are independent degree 3 polynomial features. 2.2 Correlation Structures in the Data Natural images exhibit structured dependencies in the wavelet domain: heavy–tailed marginals per coefficient, co-activa… view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of noise regimes. Clean image σ = 0 (left) to highly noised σ = 4 (right) mately orthonormal linear operator, B ∈ R p×d where p is the number of features and d = H × W is the flattened dimension of the image. B maps vectorized images Xt ∈ R d to wavelet coefficients. Noise Model We add Gaussian noise according to the variance exploding regime Xt = X0 + σZ for Z ∼ N (0, 1). We consider four no… view at source ↗
Figure 6
Figure 6. Figure 6: MNIST-64. Denoising MSE with (a) inde￾pendent features, and (b) band-tied coupling on larger images. We see higher degree polynomials are more performant with larger images, compared with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: MNIST-32. (a) Denoising MSE with inde￾pendent monomial features across three sets of features: degrees 1, 2, & 3. (b) Same features with band-tied coupling, and (c) local coupling. (d) Comparable per￾formance of a variety of trained U-Net score function approximators. (a) Independent Features (b) Band-Tied Coupling [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of linear baseline vs. wavelet models (D=3) on MNIST-32 and MNIST-64. We see the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of independent Hermite polynomials across different images sizes and degrees. Increasing [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Denoising results for the independent monomial model on two image sizes (columns) and across ranges [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Denoising results for the band-tied model on 32 by 32 images, across ranges of noise levels (groups of 3 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: MNIST samples generated by different UNet-CNN EDM diffusion model variants. Each row shows [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Generated samples for Independent vs. Band models at varying polynomial degrees. Each column [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
read the original abstract

Score-based generative models have had remarkable success over the last decade in generating a diverse set of visually plausible images. A variety of architectures including CNNs, U-Nets, and Transformers have been used as the score-approximation network in such diffusion modeling; however, to date, relatively little is known about how these architectural choices impact generative behavior. In this work, to provide insight into this area, we propose an analytically solvable parameterization of the score function using an expansion in a 2D orthogonal wavelet basis. In particular, we derive interpretable optimal score functions in terms of the moments of the data distribution. We use this parametrization to provide an architecture-agnostic, moment-based analysis that reveals which attributes of the data distribution tend to matter most for denoising. Our score machine is flexible enough to partially mimic the relevant inductive biases of multiple architectures, including U-Nets, and CNNs, taking a step towards understanding why different score architectures can exhibit distinct generative behavior. Since our score is solvable in terms of the moments of the data, we can begin to understand how the data distribution interacts with the score network to produce the behavior we observe in diffusion models.

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

1 major / 0 minor

Summary. The paper proposes an analytically solvable parameterization of the score function ∇_x log p_t(x) in diffusion models by expanding it in a 2D orthogonal wavelet basis. It claims to derive interpretable optimal score functions whose coefficients depend only on the moments of the clean data distribution, enabling an architecture-agnostic moment-based analysis of which data attributes matter most for denoising and how the parameterization can partially mimic inductive biases of U-Nets and CNNs.

Significance. If the central derivation holds, the work would offer a concrete, moment-driven lens on score networks that is independent of specific architectures, potentially explaining observed differences in generative behavior across models. The explicit link to data moments and the claim of analytical solvability would be a notable contribution to the theoretical understanding of diffusion models.

major comments (1)
  1. [Abstract / Central Claim] The central claim (abstract and introduction) that an expansion in a 2D orthogonal wavelet basis produces coefficients that are exactly functions of the (finite set of) moments of the data distribution, yielding an analytically solvable score, is not automatic from orthogonality or standard wavelet properties. The manuscript must provide the explicit derivation showing how the projection ∫ ∇_x log p_t(x) ψ_{j,k}(x) dx reduces to moment terms under the forward diffusion process; without this step the analytical solvability and moment-based analysis rest on an unproven reduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Central Claim] The central claim (abstract and introduction) that an expansion in a 2D orthogonal wavelet basis produces coefficients that are exactly functions of the (finite set of) moments of the data distribution, yielding an analytically solvable score, is not automatic from orthogonality or standard wavelet properties. The manuscript must provide the explicit derivation showing how the projection ∫ ∇_x log p_t(x) ψ_{j,k}(x) dx reduces to moment terms under the forward diffusion process; without this step the analytical solvability and moment-based analysis rest on an unproven reduction.

    Authors: We agree that the reduction from the wavelet projection of the score to explicit moment terms requires a fully explicit derivation rather than relying on standard wavelet orthogonality alone. In the revised manuscript we will insert a new subsection that starts from the closed-form expression for log p_t under the Gaussian forward process, substitutes the wavelet expansion of the score, and shows term-by-term how each coefficient integral collapses to a finite combination of raw moments of the clean data distribution via the moment-generating function of the diffusion kernel. revision: yes

Circularity Check

0 steps flagged

Wavelet expansion of score derived independently from data moments; no reduction to inputs by construction

full rationale

The paper presents a derivation of an analytically solvable score parameterization via 2D orthogonal wavelet basis expansion, with coefficients expressed as functions of data distribution moments. No load-bearing self-citations, self-definitional steps, or fitted parameters renamed as predictions appear in the provided claims. The central result is framed as following from wavelet orthogonality and diffusion forward process properties, remaining self-contained against external benchmarks without circular reduction. This matches the common honest outcome of score 0-2.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unelaborated premise that the wavelet basis permits analytical solution in terms of moments.

pith-pipeline@v0.9.1-grok · 5742 in / 1013 out tokens · 14368 ms · 2026-06-27T19:59:45.360461+00:00 · methodology

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Reference graph

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