# Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach

arXiv: https://arxiv.org/abs/2310.18677 · HTML: https://mlsanity.com/paper/2310.18677/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 65 references that are in the map.
- First posted on arXiv 28 Oct 2023.

## Builds on (most cited first, 30 of 49)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Deep Anomaly Detection with Outlier Exposure](https://mlsanity.com/paper/1812.04606/) (2018)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [DRAEM -- A discriminatively trained reconstruction embedding for surface anomaly detection](https://mlsanity.com/paper/2108.07610/) (2021)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [ViM: Out-Of-Distribution with Virtual-logit Matching](https://mlsanity.com/paper/2203.10807/) (2022)
- [Exploring the Limits of Out-of-Distribution Detection](https://mlsanity.com/paper/2106.03004/) (2021)
- [A Unified Model for Multi-class Anomaly Detection](https://mlsanity.com/paper/2206.03687/) (2022)
- [ADBench: Anomaly Detection Benchmark](https://mlsanity.com/paper/2206.09426/) (2022)
- [A Simple Fix to Mahalanobis Distance for Improving Near-OOD Detection](https://mlsanity.com/paper/2106.09022/) (2021)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [Efficient GAN-Based Anomaly Detection](https://mlsanity.com/paper/1802.06222/) (2018)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [VAEBM: A Symbiosis between Variational Autoencoders and Energy-based Models](https://mlsanity.com/paper/2010.00654/) (2020)
- [Flow Contrastive Estimation of Energy-Based Models](https://mlsanity.com/paper/1912.00589/) (2019)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)

## Built on by (most cited first, 18 of 18)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models](https://mlsanity.com/paper/2510.02300/) (2025)
- [A Survey on Diffusion Models for Anomaly Detection](https://mlsanity.com/paper/2501.11430/) (2025)
- ["Noisier" Noise Contrastive Eestimation is (Almost) Maximum Likelihood](https://mlsanity.com/paper/2405.16730/) (2024)
- [Anomaly Detection and Generation with Diffusion Models: A Survey](https://mlsanity.com/paper/2506.09368/) (2025)
- [Manifold-Constrained Energy-Based Transition Models for Offline Reinforcement Learning](https://mlsanity.com/paper/2602.02900/) (2026)
- [Hitchhiker's guide on the relation of Energy-Based Models with other generative models, sampling and statistical physics: a comprehensive review](https://mlsanity.com/paper/2406.13661/) (2024)
- [Maximum Entropy Inverse Reinforcement Learning of Diffusion Models with Energy-Based Models](https://mlsanity.com/paper/2407.00626/) (2024)
- [LSEBMCL: A Latent Space Energy-Based Model for Continual Learning](https://mlsanity.com/paper/2501.05495/) (2025)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)
- [Generalized Contrastive Divergence: Joint Training of Energy-Based Model and Diffusion Model through Inverse Reinforcement Learning](https://mlsanity.com/paper/2312.03397/) (2023)
- [Variational Potential Flow: A Novel Probabilistic Framework for Energy-Based Generative Modelling](https://mlsanity.com/paper/2407.15238/) (2024)
- [Anomaly Detection via Autoencoder Composite Features and NCE](https://mlsanity.com/paper/2502.01920/) (2025)
- [Distance Marching for Generative Modeling](https://mlsanity.com/paper/2602.02928/) (2026)
- [Deep-learning-based low-energy trigger algorithms for the Hyper-Kamiokande experiment](https://mlsanity.com/paper/2605.31391/) (2026)
- [ReFP-AD: Rectified Flow Preconditioning for Energy-Based Anomaly Detection](https://mlsanity.com/paper/2608.01793/) (2026)
- [Mitigating Convergence Collapse in Fixed-Target Anomaly Detectors via Kernel-Anchored Locality Regularization](https://mlsanity.com/paper/2610.02345/) (2026)

## Abstract (arXiv)

We present a new method of training energy-based models (EBMs) for anomaly detection that leverages low-dimensional structures within data. The proposed algorithm, Manifold Projection-Diffusion Recovery (MPDR), first perturbs a data point along a low-dimensional manifold that approximates the training dataset. Then, EBM is trained to maximize the probability of recovering the original data. The training involves the generation of negative samples via MCMC, as in conventional EBM training, but from a different distribution concentrated near the manifold. The resulting near-manifold negative samples are highly informative, reflecting relevant modes of variation in data. An energy function of MPDR effectively learns accurate boundaries of the training data distribution and excels at detecting out-of-distribution samples. Experimental results show that MPDR exhibits strong performance across various anomaly detection tasks involving diverse data types, such as images, vectors, and acoustic signals.