# Universal Smoothed Score Functions for Generative Modeling

arXiv: https://arxiv.org/abs/2303.11669 · HTML: https://mlsanity.com/paper/2303.11669/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [Deep Energy Estimator Networks](https://mlsanity.com/paper/1805.08306/) (2018)
- [Multimeasurement Generative Models](https://mlsanity.com/paper/2112.09822/) (2021)

## Built on by (most cited first, 5 of 5)

- [3D molecule generation by denoising voxel grids](https://mlsanity.com/paper/2306.07473/) (2023)
- [Structure-based drug design by denoising voxel grids](https://mlsanity.com/paper/2405.03961/) (2024)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [NEBULA: Neural Empirical Bayes Under Latent Representations for Efficient and Controllable Design of Molecular Libraries](https://mlsanity.com/paper/2407.03428/) (2024)
- [A noise-corrected Langevin algorithm and sampling by half-denoising](https://mlsanity.com/paper/2410.05837/) (2024)

## Abstract (arXiv)

We consider the problem of generative modeling based on smoothing an unknown density of interest in ℝd using factorial kernels with M independent Gaussian channels with equal noise levels introduced by Saremi and Srivastava (2022). First, we fully characterize the time complexity of learning the resulting smoothed density in ℝMd, called M-density, by deriving a universal form for its parametrization in which the score function is by construction permutation equivariant. Next, we study the time complexity of sampling an M-density by analyzing its condition number for Gaussian distributions. This spectral analysis gives a geometric insight on the "shape" of M-densities as one increases M. Finally, we present results on the sample quality in this class of generative models on the CIFAR-10 dataset where we report Fr'echet inception distances (14.15), notably obtained with a single noise level on long-run fast-mixing MCMC chains.