# CARD: Classification and Regression Diffusion Models

arXiv: https://arxiv.org/abs/2206.07275 · HTML: https://mlsanity.com/paper/2206.07275/

## Key facts

- Cited by 103 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 47 of its 85 references that are in the map.
- First posted on arXiv 15 Jun 2022.

## Placed in

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

## Builds on (most cited first, 30 of 47)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [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)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Predict Responsibly: Improving Fairness and Accuracy by Learning to Defer](https://mlsanity.com/paper/1711.06664/) (2017)
- [Concrete Dropout](https://mlsanity.com/paper/1705.07832/) (2017)

## Built on by (most cited first, 30 of 103)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [In-Context Learning Unlocked for Diffusion Models](https://mlsanity.com/paper/2305.01115/) (2023)
- [HumanMAC: Masked Motion Completion for Human Motion Prediction](https://mlsanity.com/paper/2302.03665/) (2023)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [JPEG Artifact Correction using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2209.11888/) (2022)
- [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024)
- [Enhancing Diffusion-Based Image Synthesis with Robust Classifier Guidance](https://mlsanity.com/paper/2208.08664/) (2022)
- [DiffMIC: Dual-Guidance Diffusion Network for Medical Image Classification](https://mlsanity.com/paper/2303.10610/) (2023)
- [Contrastive Conditional Latent Diffusion for Audio-visual Segmentation](https://mlsanity.com/paper/2307.16579/) (2023)
- [Spot the fake lungs: Generating Synthetic Medical Images using Neural Diffusion Models](https://mlsanity.com/paper/2211.00902/) (2022)
- [Towards Aligned Layout Generation via Diffusion Model with Aesthetic Constraints](https://mlsanity.com/paper/2402.04754/) (2024)
- [Add and Thin: Diffusion for Temporal Point Processes](https://mlsanity.com/paper/2311.01139/) (2023)
- [FedDiff: Diffusion Model Driven Federated Learning for Multi-Modal and Multi-Clients](https://mlsanity.com/paper/2401.02433/) (2023)
- [A framework for conditional diffusion modelling with applications in motif scaffolding for protein design](https://mlsanity.com/paper/2312.09236/) (2023)
- [Label-Retrieval-Augmented Diffusion Models for Learning from Noisy Labels](https://mlsanity.com/paper/2305.19518/) (2023)
- [Learning to Jump: Thinning and Thickening Latent Counts for Generative Modeling](https://mlsanity.com/paper/2305.18375/) (2023)
- [DiffusionMTL: Learning Multi-Task Denoising Diffusion Model from Partially Annotated Data](https://mlsanity.com/paper/2403.15389/) (2024)
- [DiffBFR: Bootstrapping Diffusion Model Towards Blind Face Restoration](https://mlsanity.com/paper/2305.04517/) (2023)
- [Beta Diffusion](https://mlsanity.com/paper/2309.07867/) (2023)
- [Towards Generative Modeling of Urban Flow through Knowledge-enhanced Denoising Diffusion](https://mlsanity.com/paper/2309.10547/) (2023)
- [Adding Conditional Control to Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2406.12120/) (2024)
- [NPCL: Neural Processes for Uncertainty-Aware Continual Learning](https://mlsanity.com/paper/2310.19272/) (2023)
- [MapDiffusion: Generative Diffusion for Vectorized Online HD Map Construction and Uncertainty Estimation in Autonomous Driving](https://mlsanity.com/paper/2507.21423/) (2025)
- [Annealed Multiple Choice Learning: Overcoming limitations of Winner-takes-all with annealing](https://mlsanity.com/paper/2407.15580/) (2024)
- [Diffusion Models For Multi-Modal Generative Modeling](https://mlsanity.com/paper/2407.17571/) (2024)
- [Thinking with Looped Flows](https://mlsanity.com/paper/2609.11801/) (2026)

## Abstract (arXiv)

Learning the distribution of a continuous or categorical response variable boldsymbol y given its covariates boldsymbol x is a fundamental problem in statistics and machine learning. Deep neural network-based supervised learning algorithms have made great progress in predicting the mean of boldsymbol y given boldsymbol x, but they are often criticized for their ability to accurately capture the uncertainty of their predictions. In this paper, we introduce classification and regression diffusion (CARD) models, which combine a denoising diffusion-based conditional generative model and a pre-trained conditional mean estimator, to accurately predict the distribution of boldsymbol y given boldsymbol x. We demonstrate the outstanding ability of CARD in conditional distribution prediction with both toy examples and real-world datasets, the experimental results on which show that CARD in general outperforms state-of-the-art methods, including Bayesian neural network-based ones that are designed for uncertainty estimation, especially when the conditional distribution of boldsymbol y given boldsymbol x is multi-modal. In addition, we utilize the stochastic nature of the generative model outputs to obtain a finer granularity in model confidence assessment at the instance level for classification tasks.