# Interpreting and Improving Diffusion Models from an Optimization Perspective

arXiv: https://arxiv.org/abs/2306.04848 · HTML: https://mlsanity.com/paper/2306.04848/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 37 of its 51 references that are in the map.
- First posted on arXiv 8 Jun 2023.

## Placed in

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

## Builds on (most cited first, 30 of 37)

- [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)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [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)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Zero-1-to-3: Zero-shot One Image to 3D Object](https://mlsanity.com/paper/2303.11328/) (2023)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Human Motion Diffusion Model](https://mlsanity.com/paper/2209.14916/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)

## Built on by (most cited first, 19 of 19)

- [Diffusion Policy Policy Optimization](https://mlsanity.com/paper/2409.00588/) (2024)
- [Much Ado About Noising: Dispelling the Myths of Generative Robotic Control](https://mlsanity.com/paper/2512.01809/) (2025)
- [Locality in Image Diffusion Models Emerges from Data Statistics](https://mlsanity.com/paper/2509.09672/) (2025)
- [The Duality of Generative AI and Reinforcement Learning in Robotics: A Review](https://mlsanity.com/paper/2410.16411/) (2024)
- [Accelerating Residual Reinforcement Learning with Uncertainty Estimation](https://mlsanity.com/paper/2506.17564/) (2025)
- [Any-OPD: Heterogeneous On-Policy Distillation for Flow-Matching Models via Representation-Space Bridging](https://mlsanity.com/paper/2608.03316/) (2026)
- [Enhancing Sample Generation of Diffusion Models using Noise Level Correction](https://mlsanity.com/paper/2412.05488/) (2024)
- [How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?](https://mlsanity.com/paper/2510.01404/) (2025)
- [Improving Classifier-Free Guidance of Flow Matching via Manifold Projection](https://mlsanity.com/paper/2601.21892/) (2026)
- [Delta Sampling: Data-Free Knowledge Transfer Across Diffusion Models](https://mlsanity.com/paper/2512.03056/) (2025)
- [Is Your Conditional Diffusion Model Actually Denoising?](https://mlsanity.com/paper/2512.18736/) (2025)
- [Residual Prior Diffusion: A Probabilistic Framework Integrating Coarse Latent Priors with Diffusion Models](https://mlsanity.com/paper/2512.21593/) (2025)
- [VisDiff: SDF-Guided Polygon Generation for Visibility Reconstruction and Recognition](https://mlsanity.com/paper/2410.05530/) (2024)
- [A Probabilistic Formulation of Offset Noise in Diffusion Models](https://mlsanity.com/paper/2412.03134/) (2024)
- [Distance Marching for Generative Modeling](https://mlsanity.com/paper/2602.02928/) (2026)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [Sampling sea state using a diffusion model](https://mlsanity.com/paper/2606.26389/) (2026)
- [Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers](https://mlsanity.com/paper/2607.04113/) (2026)
- [Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion](https://mlsanity.com/paper/2608.29507/) (2026)

## Abstract (arXiv)

Denoising is intuitively related to projection. Indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. Hence, learning to denoise is approximately learning to project. In this paper, we use this observation to interpret denoising diffusion models as approximate gradient descent applied to the Euclidean distance function. We then provide straight-forward convergence analysis of the DDIM sampler under simple assumptions on the projection error of the denoiser. Finally, we propose a new gradient-estimation sampler, generalizing DDIM using insights from our theoretical results. In as few as 5-10 function evaluations, our sampler achieves state-of-the-art FID scores on pretrained CIFAR-10 and CelebA models and can generate high quality samples on latent diffusion models.