# Energy-Tweedie: Score meets Score, Energy meets Energy

arXiv: https://arxiv.org/abs/2512.23818 · HTML: https://mlsanity.com/paper/2512.23818/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 43 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## Builds on (most cited first, 22 of 22)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images](https://mlsanity.com/paper/2106.07009/) (2021)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [Distributional Diffusion Models with Scoring Rules](https://mlsanity.com/paper/2502.02483/) (2025)
- [Noise Distribution Adaptive Self-Supervised Image Denoising using Tweedie Distribution and Score Matching](https://mlsanity.com/paper/2112.03696/) (2021)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [Distributional Principal Autoencoders](https://mlsanity.com/paper/2404.13649/) (2024)
- [Reverse Markov Learning: Multi-Step Generative Models for Complex Distributions](https://mlsanity.com/paper/2502.13747/) (2025)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models](https://mlsanity.com/paper/1304.3206/) (2013)
- [Whitened Score Diffusion: A Structured Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2505.10311/) (2025)
- [Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling](https://mlsanity.com/paper/2412.20045/) (2024)
- [Distributional Autoencoders Know the Score](https://mlsanity.com/paper/2502.11583/) (2025)
- [Learning a distance measure from the information-estimation geometry of data](https://mlsanity.com/paper/2510.02514/) (2025)

## Built on by (most cited first, 1 of 1)

- [Flow Mismatching: Unsupervised Anomaly Detection via Velocity Discrepancies in Flow Matching Models](https://mlsanity.com/paper/2605.23070/) (2026)

## Abstract (arXiv)

Denoising and score estimation are classically linked through Tweedie's formula, which relates the posterior mean under Gaussian noise to the Stein score of the noisy marginal. In this work, we extend this perspective beyond Gaussian noise to a broad class of Gibbs (energy-based) noise distributions, with the generalized Gaussian family as the running example. We derive the Energy-Tweedie identity: when the denoising posterior is viewed through the lens of scoring rules, the path derivative of a kernel scoring rule defined by the noise potential recovers the Stein score of the noisy marginal. The rule's propriety is determined by the noise potential alone. Thus, the familiar correspondence between Gaussian noise, posterior means, squared loss, and Tweedie's formula is lifted to a distributional correspondence between Gibbs noise distributions, full posterior laws, kernel scoring rules, and the Energy-Tweedie identity, yielding one Tweedie-style relation for each noise potential. Among its consequences, this identity gives a posterior-samples-to-score route to score estimation, yields a principled criterion for estimating unknown noise parameters, and enables diffusion-style sampling along user-chosen paths through the noise-parameter space, supplying the score-based perspective on recent generative methods trained with scoring rules.