# Hierarchical Gaussian Mixture Normalizing Flow Modeling for Unified Anomaly Detection

arXiv: https://arxiv.org/abs/2403.13349 · HTML: https://mlsanity.com/paper/2403.13349/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 50 references that are in the map.
- First posted on arXiv 20 Mar 2024.

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Towards Total Recall in Industrial Anomaly Detection](https://mlsanity.com/paper/2106.08265/) (2021)
- [PaDiM: a Patch Distribution Modeling Framework for Anomaly Detection and Localization](https://mlsanity.com/paper/2011.08785/) (2020)
- [Real-world Anomaly Detection in Surveillance Videos](https://mlsanity.com/paper/1801.04264/) (2018)
- [Unsupervised Anomaly Detection with Generative Adversarial Networks to Guide Marker Discovery](https://mlsanity.com/paper/1703.05921/) (2017)
- [DRAEM -- A discriminatively trained reconstruction embedding for surface anomaly detection](https://mlsanity.com/paper/2108.07610/) (2021)
- [Anomaly Detection via Reverse Distillation from One-Class Embedding](https://mlsanity.com/paper/2201.10703/) (2022)
- [GANomaly: Semi-Supervised Anomaly Detection via Adversarial Training](https://mlsanity.com/paper/1805.06725/) (2018)
- [SPot-the-Difference Self-Supervised Pre-training for Anomaly Detection and Segmentation](https://mlsanity.com/paper/2207.14315/) (2022)
- [Uninformed Students: Student-Teacher Anomaly Detection with Discriminative Latent Embeddings](https://mlsanity.com/paper/1911.02357/) (2019)
- [CFLOW-AD: Real-Time Unsupervised Anomaly Detection with Localization via Conditional Normalizing Flows](https://mlsanity.com/paper/2107.12571/) (2021)
- [Sub-Image Anomaly Detection with Deep Pyramid Correspondences](https://mlsanity.com/paper/2005.02357/) (2020)
- [Multiresolution Knowledge Distillation for Anomaly Detection](https://mlsanity.com/paper/2011.11108/) (2020)
- [Learning Memory-guided Normality for Anomaly Detection](https://mlsanity.com/paper/2003.13228/) (2020)
- [A Unified Model for Multi-class Anomaly Detection](https://mlsanity.com/paper/2206.03687/) (2022)
- [FastFlow: Unsupervised Anomaly Detection and Localization via 2D Normalizing Flows](https://mlsanity.com/paper/2111.07677/) (2021)
- [VT-ADL: A Vision Transformer Network for Image Anomaly Detection and Localization](https://mlsanity.com/paper/2104.10036/) (2021)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [Same Same But DifferNet: Semi-Supervised Defect Detection with Normalizing Flows](https://mlsanity.com/paper/2008.12577/) (2020)
- [The MVTec 3D-AD Dataset for Unsupervised 3D Anomaly Detection and Localization](https://mlsanity.com/paper/2112.09045/) (2021)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [Fully Convolutional Cross-Scale-Flows for Image-based Defect Detection](https://mlsanity.com/paper/2110.02855/) (2021)
- [Asymmetric Student-Teacher Networks for Industrial Anomaly Detection](https://mlsanity.com/paper/2210.07829/) (2022)
- [Towards Visually Explaining Variational Autoencoders](https://mlsanity.com/paper/1911.07389/) (2019)
- [Pixel-wise Anomaly Detection in Complex Driving Scenes](https://mlsanity.com/paper/2103.05445/) (2021)

## Built on by (most cited first, 28 of 28)

- [Towards Zero-Shot Anomaly Detection and Reasoning with Multimodal Large Language Models](https://mlsanity.com/paper/2502.07601/) (2025)
- [Exploring Intrinsic Normal Prototypes within a Single Image for Universal Anomaly Detection](https://mlsanity.com/paper/2503.02424/) (2025)
- [MANTA: A Large-Scale Multi-View and Visual-Text Anomaly Detection Dataset for Tiny Objects](https://mlsanity.com/paper/2412.04867/) (2024)
- [ResAD: A Simple Framework for Class Generalizable Anomaly Detection](https://mlsanity.com/paper/2410.20047/) (2024)
- [Salvaging the Overlooked: Leveraging Class-Aware Contrastive Learning for Multi-Class Anomaly Detection](https://mlsanity.com/paper/2412.04769/) (2024)
- [Distribution Prototype Diffusion Learning for Open-set Supervised Anomaly Detection](https://mlsanity.com/paper/2502.20981/) (2025)
- [Center-aware Residual Anomaly Synthesis for Multi-class Industrial Anomaly Detection](https://mlsanity.com/paper/2505.17551/) (2025)
- [INP-Former++: Advancing Universal Anomaly Detection via Intrinsic Normal Prototypes and Residual Learning](https://mlsanity.com/paper/2506.03660/) (2025)
- [ROADS: Robust Prompt-driven Multi-Class Anomaly Detection under Domain Shift](https://mlsanity.com/paper/2411.16049/) (2024)
- [Visual Anomaly Detection under Complex View-Illumination Interplay: A Large-Scale Benchmark](https://mlsanity.com/paper/2505.10996/) (2025)
- [Continual-MEGA: A Large-scale Benchmark for Generalizable Continual Anomaly Detection](https://mlsanity.com/paper/2506.00956/) (2025)
- [Towards an Incremental Unified Multimodal Anomaly Detection: Augmenting Multimodal Denoising From an Information Bottleneck Perspective](https://mlsanity.com/paper/2603.02629/) (2026)
- [A Comprehensive Survey for Real-World Industrial Defect Detection: Challenges, Approaches, and Prospects](https://mlsanity.com/paper/2507.13378/) (2025)
- [ResAD++: Towards Class Agnostic Anomaly Detection via Residual Feature Learning](https://mlsanity.com/paper/2509.23741/) (2025)
- [ADNet: A Large-Scale and Extensible Multi-Domain Benchmark for Anomaly Detection Across 380 Real-World Categories](https://mlsanity.com/paper/2511.20169/) (2025)
- [Toward Long-Tailed Online Anomaly Detection through Class-Agnostic Concepts](https://mlsanity.com/paper/2507.16946/) (2025)
- [Unsupervised Multi-View Visual Anomaly Detection via Progressive Homography-Guided Alignment](https://mlsanity.com/paper/2511.18766/) (2025)
- [MMR-AD: A Large-Scale Multimodal Dataset for Benchmarking General Anomaly Detection with Multimodal Large Language Models](https://mlsanity.com/paper/2604.10971/) (2026)
- [Fractal Flow: Hierarchical and Interpretable Normalizing Flow via Topic Modeling and Recursive Strategy](https://mlsanity.com/paper/2508.19750/) (2025)
- [Test-Time Adaptation for Anomaly Segmentation via Topology-Aware Optimal Transport Chaining](https://mlsanity.com/paper/2601.20333/) (2026)
- [Mixture Prototype Flow Matching for Open-Set Supervised Anomaly Detection](https://mlsanity.com/paper/2605.02438/) (2026)
- [Beyond Normal References: Discriminative Few-Shot Anomaly Detection](https://mlsanity.com/paper/2605.23231/) (2026)
- [Value-order Decomposition for Generalist Anomaly Detection](https://mlsanity.com/paper/2606.14475/) (2026)
- [DeCoFlow: Structural Decomposition of Normalizing Flows for Continual Anomaly Detection](https://mlsanity.com/paper/2606.26687/) (2026)
- [Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation](https://mlsanity.com/paper/2606.28268/) (2026)
- [ArcAD: Anomaly-Rectified Calibration for Cold-Start Supervised Anomaly Detection](https://mlsanity.com/paper/2607.02252/) (2026)
- [ReFP-AD: Rectified Flow Preconditioning for Energy-Based Anomaly Detection](https://mlsanity.com/paper/2608.01793/) (2026)
- [RagGAD: Rationale-Aware Conditional Gaussian Mixture Normalizing Flow for Unsupervised Graph Anomaly Detection](https://mlsanity.com/paper/2608.16018/) (2026)

## Abstract (arXiv)

Unified anomaly detection (AD) is one of the most challenges for anomaly detection, where one unified model is trained with normal samples from multiple classes with the objective to detect anomalies in these classes. For such a challenging task, popular normalizing flow (NF) based AD methods may fall into a "homogeneous mapping" issue,where the NF-based AD models are biased to generate similar latent representations for both normal and abnormal features, and thereby lead to a high missing rate of anomalies. In this paper, we propose a novel Hierarchical Gaussian mixture normalizing flow modeling method for accomplishing unified Anomaly Detection, which we call HGAD. Our HGAD consists of two key components: inter-class Gaussian mixture modeling and intra-class mixed class centers learning. Compared to the previous NF-based AD methods, the hierarchical Gaussian mixture modeling approach can bring stronger representation capability to the latent space of normalizing flows, so that even complex multi-class distribution can be well represented and learned in the latent space. In this way, we can avoid mapping different class distributions into the same single Gaussian prior, thus effectively avoiding or mitigating the "homogeneous mapping" issue. We further indicate that the more distinguishable different class centers, the more conducive to avoiding the bias issue. Thus, we further propose a mutual information maximization loss for better structuring the latent feature space. We evaluate our method on four real-world AD benchmarks, where we can significantly improve the previous NF-based AD methods and also outperform the SOTA unified AD methods.