# Closed-Form Diffusion Models

arXiv: https://arxiv.org/abs/2310.12395 · HTML: https://mlsanity.com/paper/2310.12395/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 43 of its 63 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Placed in

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

## Builds on (most cited first, 30 of 43)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Billion-scale similarity search with GPUs](https://mlsanity.com/paper/1702.08734/) (2017)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Novel View Synthesis with Diffusion Models](https://mlsanity.com/paper/2210.04628/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)

## Built on by (most cited first, 30 of 46)

- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Towards a Mechanistic Explanation of Diffusion Model Generalization](https://mlsanity.com/paper/2411.19339/) (2024)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Locality in Image Diffusion Models Emerges from Data Statistics](https://mlsanity.com/paper/2509.09672/) (2025)
- [Minimax Optimality of the Probability Flow ODE for Diffusion Models](https://mlsanity.com/paper/2503.09583/) (2025)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Selective Underfitting in Diffusion Models](https://mlsanity.com/paper/2510.01378/) (2025)
- [Plug-and-Play image restoration with Stochastic deNOising REgularization](https://mlsanity.com/paper/2402.01779/) (2024)
- [Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusion](https://mlsanity.com/paper/2502.21278/) (2025)
- [The Hidden Linear Structure in Score-Based Models and its Application](https://mlsanity.com/paper/2311.10892/) (2023)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [Flow Map Distillation Without Data](https://mlsanity.com/paper/2511.19428/) (2025)
- [Density Ratio Estimation with Conditional Probability Paths](https://mlsanity.com/paper/2502.02300/) (2025)
- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces](https://mlsanity.com/paper/2505.02508/) (2025)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning](https://mlsanity.com/paper/2602.18428/) (2026)
- [Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models](https://mlsanity.com/paper/2506.10177/) (2025)
- [A Kinetic Energy Perspective of Flow Matching](https://mlsanity.com/paper/2602.07928/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Momentum Guidance: Plug-and-Play Guidance for Flow Models](https://mlsanity.com/paper/2602.20360/) (2026)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [An exact information theory of generalization phase transitions in Bayesian diffusion models](https://mlsanity.com/paper/2607.08041/) (2026)
- [Bigger Isn't Always Better: Towards a General Prior for Medical Image Reconstruction](https://mlsanity.com/paper/2501.07376/) (2025)
- [Transport Novelty Distance: A Distributional Metric for Evaluating Material Generative Models](https://mlsanity.com/paper/2512.09514/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)

## Abstract (arXiv)

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.