# Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Study

arXiv: https://arxiv.org/abs/2505.22841 · HTML: https://mlsanity.com/paper/2505.22841/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 30 of its 38 references that are in the map.
- First posted on arXiv 28 May 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 30)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Understanding and Mitigating Copying in Diffusion Models](https://mlsanity.com/paper/2305.20086/) (2023)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Accelerating Diffusion Models via Early Stop of the Diffusion Process](https://mlsanity.com/paper/2205.12524/) (2022)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [On Memorization in Probabilistic Deep Generative Models](https://mlsanity.com/paper/2106.03216/) (2021)
- [Towards Memorization-Free Diffusion Models](https://mlsanity.com/paper/2404.00922/) (2024)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure](https://mlsanity.com/paper/2410.24060/) (2024)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [On the Generalization of Diffusion Model](https://mlsanity.com/paper/2305.14712/) (2023)
- [An analysis of the noise schedule for score-based generative models](https://mlsanity.com/paper/2402.04650/) (2024)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [A solvable model of learning generative diffusion: theory and insights](https://mlsanity.com/paper/2501.03937/) (2025)
- [Shallow diffusion networks provably learn hidden low-dimensional structure](https://mlsanity.com/paper/2410.11275/) (2024)
- [Analysis of Diffusion Models for Manifold Data](https://mlsanity.com/paper/2502.04339/) (2025)

## Built on by (most cited first, 3 of 3)

- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Optimal score function estimation via derivatives constraints](https://mlsanity.com/paper/2606.19084/) (2026)
- [Training-Free Generative Sampling via Moment-Matched Score Smoothing](https://mlsanity.com/paper/2605.14276/) (2026)

## Abstract (arXiv)

Diffusion models now set the benchmark in high-fidelity generative sampling, yet they can, in principle, be prone to memorization. In this case, their learned score overfits the finite dataset so that the reverse-time SDE samples are mostly training points. In this paper, we interpret the empirical score as a noisy version of the true score and show that its covariance matrix is asymptotically a re-weighted data PCA. In large dimension, the small time limit makes the noise variance blow up while simultaneously reducing spatial correlation. To reduce this variance, we introduce a kernel-smoothed empirical score and analyze its bias-variance trade-off. We derive asymptotic bounds on the Kullback-Leibler divergence between the true distribution and the one generated by the modified reverse SDE. Regularization on the score has the same effect as increasing the size of the training dataset, and thus helps prevent memorization. A spectral decomposition of the forward diffusion suggests better variance control under some regularity conditions of the true data distribution. Reverse diffusion with kernel-smoothed empirical score can be reformulated as a gradient descent drifted toward a Log-Exponential Double-Kernel Density Estimator (LED-KDE). This perspective highlights two regularization mechanisms taking place in denoising diffusions: an initial Gaussian kernel first diffuses mass isotropically in the ambient space, while a second kernel applied in score space concentrates and spreads that mass along the data manifold. Hence, even a straightforward regularization-without any learning-already mitigates memorization and enhances generalization. Numerically, we illustrate our results with several experiments on synthetic and MNIST datasets.