# High-dimensional and Permutation Invariant Anomaly Detection

arXiv: https://arxiv.org/abs/2306.03933 · HTML: https://mlsanity.com/paper/2306.03933/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 104 references that are in the map.
- First posted on arXiv 6 Jun 2023.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [A Living Review of Machine Learning for Particle Physics](https://mlsanity.com/paper/2102.02770/) (2021)
- [Anomaly Detection with Density Estimation](https://mlsanity.com/paper/2001.04990/) (2020)
- [Score-based Generative Models for Calorimeter Shower Simulation](https://mlsanity.com/paper/2206.11898/) (2022)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [CaloFlow: Fast and Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2106.05285/) (2021)
- [Classification without labels: Learning from mixed samples in high energy physics](https://mlsanity.com/paper/1708.02949/) (2017)
- [CaloFlow II: Even Faster and Still Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2110.11377/) (2021)
- [CaloClouds: Fast Geometry-Independent Highly-Granular Calorimeter Simulation](https://mlsanity.com/paper/2305.04847/) (2023)
- [The Dark Machines Anomaly Score Challenge: Benchmark Data and Model Independent Event Classification for the Large Hadron Collider](https://mlsanity.com/paper/2105.14027/) (2021)
- [PC-JeDi: Diffusion for Particle Cloud Generation in High Energy Physics](https://mlsanity.com/paper/2303.05376/) (2023)
- [DCTRGAN: Improving the Precision of Generative Models with Reweighting](https://mlsanity.com/paper/2009.03796/) (2020)
- [Inductive Simulation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2305.11934/) (2023)
- [Novelty Detection Meets Collider Physics](https://mlsanity.com/paper/1807.10261/) (2018)
- [JUNIPR: a Framework for Unsupervised Machine Learning in Particle Physics](https://mlsanity.com/paper/1804.09720/) (2018)
- [Comparing Weak- and Unsupervised Methods for Resonant Anomaly Detection](https://mlsanity.com/paper/2104.02092/) (2021)
- [Anomalies, Representations, and Self-Supervision](https://mlsanity.com/paper/2301.04660/) (2023)
- [Latent Space Refinement for Deep Generative Models](https://mlsanity.com/paper/2106.00792/) (2021)
- [IRC-safe Graph Autoencoder for unsupervised anomaly detection](https://mlsanity.com/paper/2204.12231/) (2022)
- [ELSA -- Enhanced latent spaces for improved collider simulations](https://mlsanity.com/paper/2305.07696/) (2023)
- [Weakly-Supervised Anomaly Detection in the Milky Way](https://mlsanity.com/paper/2305.03761/) (2023)

## Built on by (most cited first, 12 of 12)

- [Machine Learning for Anomaly Detection in Particle Physics](https://mlsanity.com/paper/2312.14190/) (2023)
- [Deep Generative Models for Detector Signature Simulation: A Taxonomic Review](https://mlsanity.com/paper/2312.09597/) (2023)
- [Improving new physics searches with diffusion models for event observables and jet constituents](https://mlsanity.com/paper/2312.10130/) (2023)
- [PC-Droid: Faster diffusion and improved quality for particle cloud generation](https://mlsanity.com/paper/2307.06836/) (2023)
- [Full Event Particle-Level Unfolding with Variable-Length Latent Variational Diffusion](https://mlsanity.com/paper/2404.14332/) (2024)
- [Anomaly Detection and Generation with Diffusion Models: A Survey](https://mlsanity.com/paper/2506.09368/) (2025)
- [Unifying Simulation and Inference with Normalizing Flows](https://mlsanity.com/paper/2404.18992/) (2024)
- [Model-Agnostic Signal Discovery with Machine Learning: Bridging the Gap Between Theory and Practice](https://mlsanity.com/paper/2605.31103/) (2026)
- [Data-Driven High-Dimensional Statistical Inference with Generative Models](https://mlsanity.com/paper/2506.06438/) (2025)
- [Variational inference for pile-up removal at hadron colliders with diffusion models](https://mlsanity.com/paper/2410.22074/) (2024)
- [TRANSIT your events into a new mass: Fast background interpolation for weakly-supervised anomaly searches](https://mlsanity.com/paper/2503.04342/) (2025)
- [Contrastive Learning for Interpretable Anomaly Detection at Collider Experiments](https://mlsanity.com/paper/2608.13652/) (2026)

## Abstract (arXiv)

Methods for anomaly detection of new physics processes are often limited to low-dimensional spaces due to the difficulty of learning high-dimensional probability densities. Particularly at the constituent level, incorporating desirable properties such as permutation invariance and variable-length inputs becomes difficult within popular density estimation methods. In this work, we introduce a permutation-invariant density estimator for particle physics data based on diffusion models, specifically designed to handle variable-length inputs. We demonstrate the efficacy of our methodology by utilizing the learned density as a permutation-invariant anomaly detection score, effectively identifying jets with low likelihood under the background-only hypothesis. To validate our density estimation method, we investigate the ratio of learned densities and compare to those obtained by a supervised classification algorithm.