# Heteroscedasticity of Denoising Score Matching with Generalised Smooth Noise

arXiv: https://arxiv.org/abs/2508.01597 · HTML: https://mlsanity.com/paper/2508.01597/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 28 references that are in the map.
- First posted on arXiv 3 Aug 2025.

## Builds on (most cited first, 19 of 19)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)
- [GENIE: Higher-Order Denoising Diffusion Solvers](https://mlsanity.com/paper/2210.05475/) (2022)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Stable Target Field for Reduced Variance Score Estimation in Diffusion Models](https://mlsanity.com/paper/2302.00670/) (2023)
- [Heavy-Tailed Diffusion Models](https://mlsanity.com/paper/2410.14171/) (2024)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Heavy-tailed denoising score matching](https://mlsanity.com/paper/2112.09788/) (2021)
- [One Size Fits All: Can We Train One Denoiser for All Noise Levels?](https://mlsanity.com/paper/2005.09627/) (2020)
- [Loss Functions in Diffusion Models: A Comparative Study](https://mlsanity.com/paper/2507.01516/) (2025)

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

- [Factor-Based Conditional Diffusion Model for Contextual Portfolio Optimization](https://mlsanity.com/paper/2509.22088/) (2025)

## Abstract (arXiv)

Score Matching (SM) is a powerful framework for estimating the log-density derivatives of a distribution without calculating its normalizing constants. This capability has made it a cornerstone across multiple domains, from classical sta- tistical estimation and energy-based models to modern diffusion-based generative models. In practice, these models rely almost exclusively on Denoising Score Matching (DSM) as a tractable proxy for score matching. This ubiquity naturally raises a fundamental question: Is DSM truly "score matching for free"? In this work, we demonstrate that DSM is not a perfect substitute. We prove that the denoising objective is inherently heteroscedastic, the variance of model parame- ters fluctuates unpredictably based on both noise levels and the underlying data geometry. This instability is baked into the mathematical structure of the DSM. To address this, we derive an ideal weighting function that equalizes this variance, yielding a homoscedastic generalization of DSM. Since the ideal weights are of- ten empirically inaccessible, we show that a practical approximation weighting function via Taylor expansion reduces gradient variance during training, at the cost of statistical optimality. Notably, this provides a theoretical justification for an existing heuristic weight used in Isotropic Gaussian Diffusion. We validate our theory across different perturbed distributions and for higher-order scores.