# E Pluribus Unum Ex Machina: Learning from Many Collider Events at Once arXiv: https://arxiv.org/abs/2101.07263 ยท HTML: https://mlsanity.com/paper/2101.07263/ ## Key facts - Cited by 15 papers in the map (data through 14 Oct 2026). - Builds on 22 of its 71 references that are in the map. - First posted on arXiv 18 Jan 2021. ## Builds on (most cited first, 22 of 22) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013) - [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016) - [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015) - [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017) - [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020) - [Approximating Likelihood Ratios with Calibrated Discriminative Classifiers](https://mlsanity.com/paper/1506.02169/) (2015) - [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018) - [Anomaly Detection with Density Estimation](https://mlsanity.com/paper/2001.04990/) (2020) - [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018) - [Machine Learning in High Energy Physics Community White Paper](https://mlsanity.com/paper/1807.02876/) (2018) - [Classification without labels: Learning from mixed samples in high energy physics](https://mlsanity.com/paper/1708.02949/) (2017) - [Parameterized Machine Learning for High-Energy Physics](https://mlsanity.com/paper/1601.07913/) (2016) - [MadMiner: Machine learning-based inference for particle physics](https://mlsanity.com/paper/1907.10621/) (2019) - [OmniFold: A Method to Simultaneously Unfold All Observables](https://mlsanity.com/paper/1911.09107/) (2019) - [Simulation Assisted Likelihood-free Anomaly Detection](https://mlsanity.com/paper/2001.05001/) (2020) - [Constraining Effective Field Theories with Machine Learning](https://mlsanity.com/paper/1805.00013/) (2018) - [A Guide to Constraining Effective Field Theories with Machine Learning](https://mlsanity.com/paper/1805.00020/) (2018) - [Neural Networks for Full Phase-space Reweighting and Parameter Tuning](https://mlsanity.com/paper/1907.08209/) (2019) - [JUNIPR: a Framework for Unsupervised Machine Learning in Particle Physics](https://mlsanity.com/paper/1804.09720/) (2018) - [Likelihood-free inference with an improved cross-entropy estimator](https://mlsanity.com/paper/1808.00973/) (2018) - [Parameter Estimation using Neural Networks in the Presence of Detector Effects](https://mlsanity.com/paper/2010.03569/) (2020) ## Built on by (most cited first, 15 of 15) - [A Living Review of Machine Learning for Particle Physics](https://mlsanity.com/paper/2102.02770/) (2021) - [CaloFlow: Fast and Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2106.05285/) (2021) - [New directions for surrogate models and differentiable programming for High Energy Physics detector simulation](https://mlsanity.com/paper/2203.08806/) (2022) - [Learning Likelihood Ratios with Neural Network Classifiers](https://mlsanity.com/paper/2305.10500/) (2023) - [Latent Space Refinement for Deep Generative Models](https://mlsanity.com/paper/2106.00792/) (2021) - [Hierarchical Neural Simulation-Based Inference Over Event Ensembles](https://mlsanity.com/paper/2306.12584/) (2023) - [Resonant Anomaly Detection with Multiple Reference Datasets](https://mlsanity.com/paper/2212.10579/) (2022) - [Bias and Priors in Machine Learning Calibrations for High Energy Physics](https://mlsanity.com/paper/2205.05084/) (2022) - [Machine-Learned Exclusion Limits without Binning](https://mlsanity.com/paper/2211.04806/) (2022) - [Neural Quasiprobabilistic Likelihood Ratio Estimation with Negatively Weighted Data](https://mlsanity.com/paper/2410.10216/) (2024) - [Towards a method to anticipate dark matter signals with deep learning at the LHC](https://mlsanity.com/paper/2105.12018/) (2021) - [Discriminative versus Generative Approaches to Simulation-based Inference](https://mlsanity.com/paper/2503.07962/) (2025) - [BitHEP -- The Limits of Low-Precision ML in HEP](https://mlsanity.com/paper/2504.03387/) (2025) - [Increasing Information Extraction in Low-Signal Regimes via Multiple Instance Learning](https://mlsanity.com/paper/2508.07114/) (2025) - [Localizing Global Discrepancies: Marginal Contributions and Contextual Anomaly Detection](https://mlsanity.com/paper/2608.28375/) (2026) ## Abstract (arXiv) There have been a number of recent proposals to enhance the performance of machine learning strategies for collider physics by combining many distinct events into a single ensemble feature. To evaluate the efficacy of these proposals, we study the connection between single-event classifiers and multi-event classifiers under the assumption that collider events are independent and identically distributed (IID). We show how one can build optimal multi-event classifiers from single-event classifiers, and we also show how to construct multi-event classifiers such that they produce optimal single-event classifiers. This is illustrated for a Gaussian example as well as for classification tasks relevant for searches and measurements at the Large Hadron Collider. We extend our discussion to regression tasks by showing how they can be phrased in terms of parametrized classifiers. Empirically, we find that training a single-event (per-instance) classifier is more effective than training a multi-event (per-ensemble) classifier, as least for the cases we studied, and we relate this fact to properties of the loss function gradient in the two cases. While we did not identify a clear benefit from using multi-event classifiers in the collider context, we speculate on the potential value of these methods in cases involving only approximate independence, as relevant for jet substructure studies.