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arxiv: 2605.15487 · v1 · pith:3KZUDKN7new · submitted 2026-05-15 · 💻 cs.LG · cs.CV· eess.IV

Learning Normalized Energy Models for Linear Inverse Problems

Nicolas Zilberstein , Santiago Segarra , Eero Simoncelli , Florentin Guth This is my paper

Pith reviewed 2026-05-20 21:18 UTC · model grok-4.3

classification 💻 cs.LG cs.CVeess.IV
keywords energy-based modelsdiffusion modelslinear inverse problemsdenoisingnormalized posteriorsimaginginpaintingdeblurring
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The pith

Energy model trained only on denoising computes normalized posteriors for any linear inverse problem without retraining

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

The paper presents an energy-based model trained solely for denoising, augmented by a covariance-based regularization term that enforces consistency across varying measurement conditions. This design allows the model to directly evaluate normalized posterior densities for a range of linear inverse problems such as inpainting and deblurring. By making the prior density explicit rather than implicit, the approach retains the sampling strengths of diffusion models while enabling new operations including on-the-fly adaptive sampling schedules, unbiased Metropolis-Hastings corrections, and blind estimation of the degradation operator. The method is validated on standard image datasets and tasks, showing performance that matches or exceeds existing baselines.

Core claim

We introduce a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model can compute normalized posterior densities for diverse linear inverse problems, without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule.

What carries the argument

covariance-based regularization term that enforces consistency across different measurement conditions in an energy-based model trained for denoising

If this is right

  • Energy-guided adaptive sampling can adjust schedules on-the-fly during inference for linear inverse problems.
  • Unbiased Metropolis-Hastings correction steps become available because normalized densities are explicitly computable.
  • Blind estimation of the degradation operator is possible by applying Bayes rule to the normalized posterior.
  • The same trained model applies directly to inpainting and deblurring tasks on datasets such as ImageNet and CelebA without fine-tuning.

Where Pith is reading between the lines

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

  • The explicit normalized densities could support uncertainty quantification in downstream imaging pipelines that currently rely on heuristic scores.
  • If the consistency property holds for linear operators, analogous regularization might be designed for selected classes of nonlinear degradations to broaden the method's scope.
  • The approach suggests a route to replace implicit priors in other generative frameworks with explicit, normalizable energy functions when consistency across conditions can be enforced.

Load-bearing premise

The covariance-based regularization term enforces consistency across different measurement conditions so that a model trained only on denoising generalizes directly to compute normalized posteriors for arbitrary linear inverse problems.

What would settle it

A direct test would compare the model's computed posterior density values against exact normalized densities obtained by exhaustive enumeration or high-precision sampling on a simple linear inverse problem such as one-dimensional deconvolution; large systematic deviations would falsify the generalization claim.

Figures

Figures reproduced from arXiv: 2605.15487 by Eero Simoncelli, Florentin Guth, Nicolas Zilberstein, Santiago Segarra.

Figure 1
Figure 1. Figure 1: Illustration of possible paths in image space (left) and covariance space (right). Isotropic models are limited to covariance schedules that are reparameterizations of the diagonal (orange), while anisotropic models can explore paths where different signal components are noise-corrupted at different rates (red, black, and purple). papers (Elata et al., 2025; Terris et al., 2026) have proposed degradation-a… view at source ↗
Figure 2
Figure 2. Figure 2: Proposed architecture based on UNet. We incorporate the covariance information through the embedding network, with two dedicated branches for the two covariance domains (spatial and spectral). horizontal-box masks with sizes from 1 to d 2 = 642 . In the spectral domain, we consider Gaussian deblurring and ×4 super-resolution. Additional details regarding hyper￾parameters and examples of the different covar… view at source ↗
Figure 4
Figure 4. Figure 4: Left and middle: Histograms of log pθ(xˆ) and log pθ(xˆ|y) for inpainting solutions xˆ generated by DPS, RED-Diff, and our energy model from a given measurement y, along with the ground truth x. Our energy model is well-calibrated with respect to both prior and posterior probabilities. Right: Examples of generated images xˆ sorted from lowest to highest prior probability (in reading order). The colored bor… view at source ↗
Figure 5
Figure 5. Figure 5: Three autoregressive generations for a model trained on MNIST, with identical initial and injected noise. The first two rows generate the four quadrants of the image respectively in reading and reversed reading order. The third row generates the image in 16 patches, ordered from the center to the outer border. task of reconstructing MNIST digits from a subset of k ran￾domly selected pixels (unobserved pixe… view at source ↗
Figure 7
Figure 7. Figure 7: Blind reconstruction experiment. Top: two observations corrupted by noise of standard deviation σ1 inside a central s × s square, with (σ1, s) ∈ {(0.1, 14),(2, 20)}). Middle and bottom: Log probability log pθ(y|Σs) as a function of box size s and σ1, respectively. The noise covariance parameters estimated by the energy model trained with dual score matching (red dashed vertical lines) accurately track thei… view at source ↗
Figure 8
Figure 8. Figure 8: Examples of particular Σt for different kernels and masks. 0 5 10 15 20 25 X 0 5 10 15 20 25 Y num_boxes=1 box_size=28x28 0 5 10 15 20 25 X 0 5 10 15 20 25 Y num_boxes=4 box_size=14x14 0 5 10 15 20 25 X 0 5 10 15 20 25 Y num_boxes=16 box_size=7x7 0 5 10 15 20 25 X 0 5 10 15 20 25 Y num_boxes=49 box_size=4x4 0 5 10 15 20 25 X 0 5 10 15 20 25 Y num_boxes=196 box_size=2x2 0 5 10 15 20 25 X 0 5 10 15 20 25 Y n… view at source ↗
Figure 9
Figure 9. Figure 9: Examples of autoregressive patch-based covariances Σt used for training. We first sample the number of boxes, and then the variance of each box. C.3. Comparison with previous energy-based models Our method is more computationally intensive to train than standard score-based models. This overhead primarily comes from double backpropagation, which is inherent to energy-based formulations and is widely acknow… view at source ↗
Figure 10
Figure 10. Figure 10: Noisy observation with a box covariance of size 30 × 30 and σ1 = 1. 20 25 30 35 40 45 50 Box size s 10.5 11.0 11.5 12.0 12.5 13.0 lo g p (x) [d B / dim] Dual score matching Ground-truth 0 20 40 60 80 100 1 2 4 6 8 10 lo g p (x) [d B / dim] Dual score matching Ground-truth [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Blind reconstruction experiment. Log probability log pθ(y|Σs) as a function of box size s and σ1, respectively. The energy model trained with dual score matching correctly identifies the true box size (dashed red for estimated size and solid green vertical lines). D.2. Comparison between dual and single score matching Blind experiment. Validating the correctness of the dual score matching is difficult sin… view at source ↗
Figure 12
Figure 12. Figure 12: Log probability log pθ(y|Σs) as a function of box size s for s = {5, 14, 20} with σ1 = {2, 0.1, 2} respectively. The energy model trained with dual score matching correctly identifies the true box size (dashed red for estimated size and solid green vertical lines), which is not the case for the model trained with single score matching (dashed blue vertical line). One-shot conditional MMSE estimator. One a… view at source ↗
Figure 13
Figure 13. Figure 13: MMSE for one-shot denoising in box inpainting (10 × 10) as a function of the noise in the box. Noisy images MMSE estimator [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MMSE estimator computed by our model, which corresponds to xˆ = E[x|y, Σ]. D.3. Analysis of the posterior distribution CelebA. We include here additional results for the analysis of the posterior distribution described in Section 4.1. We start showing the distribution across multiple (x, y) in [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Histograms of log pθ(xˆ) and log pθ(xˆ|y) for inpainting solutions xˆ generated by DPS, RED-Diff, and our energy model for different measurements y associated to different x, along with the ground truths x. Our energy model is well-calibrated with respect to both prior and posterior probabilities. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Images sorted from high to low probability (from left-up corner to right-down corner). Images with highest prior probability contains less details than those associated with higer probability. ImageNet. We evaluate our model on ImageNet64, first validating that the learned prior aligns with the complexity￾probability relationship described in Guth et al. (2025). As illustrated in [PITH_FULL_IMAGE:figures… view at source ↗
Figure 17
Figure 17. Figure 17: ImageNet posterior samples for a 21 × 21 center inpainting task, sorted by decreasing prior probability. Consistent with previous observations, samples with the highest probability feature uniform backgrounds and minimal detail, whereas lower-probability samples exhibit significantly higher structural complexity and dense detail [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: ImageNet posterior samples (red border) generated by sampling from measurements y generated using a box inpainting mask with a box of size 21 × 21, and sorted from high to low posterior probability, next to their corresponding ground truth (blue border). D.4. Recovering the Gumbel distribution for the isotropic case Our model generalizes the energy model of (Guth et al., 2025), which is recovered as a spe… view at source ↗
Figure 19
Figure 19. Figure 19: Histogram of log probabilities log pθ(x)[dB/dim] for 50k in the ImageNet test set. The distribution is well-fit by a Gumbel distribution (red line), but there is a second mode (centered near 38 dB/dim) in contrast to the model in [1]. This additional mode is caused by grayscale images within ImageNet; these samples possess significantly higher probability under the model Uθ compared to standard RGB sample… view at source ↗
Figure 20
Figure 20. Figure 20: Images from the test set of ImageNet 64 × 64, sorted from low to high probability (left to right and top to down). D.5. Validation of the normalization constant in synthethic experiment Analogous to the experiments of (Guth et al., 2025), we generate n = 100,000 samples from a mixture of two Gaussian distributions, 1 2N (0, σ2 1 Id) + 1 2N (0, σ2 2 Id), with σ1 = 1 and σ2 = 4, in dimension d = 1,000. Each… view at source ↗
Figure 21
Figure 21. Figure 21: Comparison of single and dual score matching on a 1000-dimensional Gaussian scale mixture. Left: Dual score matching (red dashed) captures the true energy (blue solid), whereas single score matching (green) fails, even after global normalization. Right: Radial score components. Single score matching learns the score accurately within the data support (blue bar plot) but lacks accuracy outside of this regi… view at source ↗
Figure 22
Figure 22. Figure 22: Examples of inpainting for CelebA. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Examples of inpainting for ImageNet. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Examples of deblurring for CelebA 31 [PITH_FULL_IMAGE:figures/full_fig_p031_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Examples of deblurring for ImageNet 32 [PITH_FULL_IMAGE:figures/full_fig_p032_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Examples of super-resolution for CelebA 33 [PITH_FULL_IMAGE:figures/full_fig_p033_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Examples of inpainting for AFHQ-cats. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Examples of Motion Deblurring. MALA ULA Energy-guided 1 5 8 Steps [PITH_FULL_IMAGE:figures/full_fig_p035_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Comparison of the different corrector samplers a function of the number of steps. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_29.png] view at source ↗
read the original abstract

Generative diffusion models can provide powerful prior probability models for inverse problems in imaging, but existing implementations suffer from two key limitations: $(i)$ the prior density is represented implicitly, and $(ii)$ they rely on likelihood approximations that introduce sampling biases. We address these challenges by introducing a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model can compute normalized posterior densities for diverse linear inverse problems, without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule. We validate the method on multiple datasets (ImageNet, CelebA, AFHQ) and tasks (inpainting, deblurring), demonstrating competitive or superior performance to established baselines.

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 / 1 minor

Summary. The paper introduces a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model is claimed to compute normalized posterior densities for diverse linear inverse problems without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule. The method is validated on ImageNet, CelebA, and AFHQ datasets for inpainting and deblurring tasks, demonstrating competitive or superior performance to established baselines.

Significance. If the central claim holds, this would represent a meaningful advance in applying generative models to inverse problems by enabling exact normalized posterior computation from a denoising-trained model, which could support new exact inference techniques such as unbiased corrections and adaptive sampling while retaining diffusion-style generation. The generalization from denoising to arbitrary linear operators via the regularizer, if rigorously established, would be a notable technical contribution.

major comments (1)
  1. [Methods (training objective and regularization)] The central claim that the covariance-based regularization produces energies whose normalization constants yield exact posterior densities p(x|y) for arbitrary linear degradations H rests on the regularizer enforcing that E_θ(x, y) differs from -log p(x|y) by a y-dependent constant independent of x. No derivation is provided showing this property holds for operators outside the denoising training distribution (e.g., structured blur), and if the regularizer only matches second moments under the training noise, the partition function for a new H can still depend on x, breaking the normalization guarantee.
minor comments (1)
  1. [Experiments] The abstract and validation description mention competitive performance on inpainting and deblurring but would benefit from explicit quantitative metrics, baseline comparisons, and controls for the new capabilities (e.g., adaptive sampling) in the results section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. The major comment raises an important point about the theoretical grounding of the normalization property. We address it directly below and will revise the manuscript to include additional derivations and clarifications.

read point-by-point responses

  1. Referee: The central claim that the covariance-based regularization produces energies whose normalization constants yield exact posterior densities p(x|y) for arbitrary linear degradations H rests on the regularizer enforcing that E_θ(x, y) differs from -log p(x|y) by a y-dependent constant independent of x. No derivation is provided showing this property holds for operators outside the denoising training distribution (e.g., structured blur), and if the regularizer only matches second moments under the training noise, the partition function for a new H can still depend on x, breaking the normalization guarantee.

    Authors: We appreciate the referee's precise articulation of the required property. The covariance regularization is designed to enforce consistency of the learned energy across measurement conditions by penalizing mismatches in the model's predicted covariances, which under the linear-Gaussian measurement model ensures that any x-dependent terms in the partition function cancel. This is motivated in Section 3.2 and the supplementary material, where we show that matching second-order statistics across perturbed y's yields an energy whose normalizing constant depends only on y. However, we agree that an explicit step-by-step derivation extending the argument from isotropic denoising noise to general structured linear operators H (such as non-uniform blurs) is not fully expanded in the main text. In the revised version we will add a self-contained proof in the appendix that starts from the regularized objective and shows Z(y,H) is independent of x for any fixed linear H, under the assumption that the training distribution covers the requisite second-moment statistics. This addition will also include a brief discussion of the conditions under which the property may degrade for highly structured degradations far from the training distribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with explicit regularization term

full rationale

The paper trains an energy-based model on denoising using an explicit covariance-based regularization term to promote consistency across measurement conditions. This objective is stated separately from the downstream claim of normalized posteriors for arbitrary linear operators H. No equation reduces a claimed prediction to a fitted quantity by construction, no self-citation chain is load-bearing for the normalization property, and no ansatz is smuggled via prior work. Validation on inpainting and deblurring tasks provides independent empirical support rather than tautological redefinition. The central result therefore rests on the proposed loss and its generalization behavior, not on re-expressing inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the covariance regularization is presented as a training choice without further breakdown.

pith-pipeline@v0.9.0 · 5694 in / 1100 out tokens · 48742 ms · 2026-05-20T21:18:47.429814+00:00 · methodology

discussion (0)

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