# Random-Set Neural Networks (RS-NN)

arXiv: https://arxiv.org/abs/2307.05772 · HTML: https://mlsanity.com/paper/2307.05772/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 151 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Builds on (most cited first, 30 of 57)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [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)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Revisiting the Calibration of Modern Neural Networks](https://mlsanity.com/paper/2106.07998/) (2021)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Posterior Network: Uncertainty Estimation without OOD Samples via Density-Based Pseudo-Counts](https://mlsanity.com/paper/2006.09239/) (2020)
- [Bayesian Convolutional Neural Networks with Bernoulli Approximate Variational Inference](https://mlsanity.com/paper/1506.02158/) (2015)
- [Ensemble Distribution Distillation](https://mlsanity.com/paper/1905.00076/) (2019)
- [Uncertainty Quantification and Deep Ensembles](https://mlsanity.com/paper/2007.08792/) (2020)

## Built on by (most cited first, 19 of 19)

- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [A Unified Evaluation Framework for Epistemic Predictions](https://mlsanity.com/paper/2501.16912/) (2025)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Random-Set Large Language Models](https://mlsanity.com/paper/2504.18085/) (2025)
- [Learning Credal Ensembles via Distributionally Robust Optimization](https://mlsanity.com/paper/2602.08470/) (2026)
- [Reasoning with random sets: An agenda for the future](https://mlsanity.com/paper/2401.09435/) (2023)
- [Generalising realisability in statistical learning theory under epistemic uncertainty](https://mlsanity.com/paper/2402.14759/) (2024)
- [Epistemic Wrapping for Uncertainty Quantification](https://mlsanity.com/paper/2505.02277/) (2025)
- [Uncertainty-Aware Autonomous Vehicles: Predicting the Road Ahead](https://mlsanity.com/paper/2510.22680/) (2025)
- [Epistemic Generative Adversarial Networks](https://mlsanity.com/paper/2603.18348/) (2026)
- [Random-Set Graph Neural Networks](https://mlsanity.com/paper/2605.11987/) (2026)
- [Credal Large Language Models for Semantic Commitment under Uncertainty](https://mlsanity.com/paper/2608.23244/) (2026)
- [Set-based v.s. Distribution-based Representations of Epistemic Uncertainty: A Comparative Study](https://mlsanity.com/paper/2602.22747/) (2026)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)
- [A formal definition and meta-model for a machine theory of mind](https://mlsanity.com/paper/2606.03471/) (2026)
- [What Intermediate Layers Know: Detecting Jailbreaks from Entropy Dynamics](https://mlsanity.com/paper/2606.25182/) (2026)
- [Decision-Aligned Evaluation of Uncertainty Quantification](https://mlsanity.com/paper/2606.26990/) (2026)
- [Possibilistic Predictive Uncertainty for Deep Learning](https://mlsanity.com/paper/2605.00600/) (2026)
- [XU-RS: Explaining Credal Width in Random-Set Language Models](https://mlsanity.com/paper/2609.37594/) (2026)

## Abstract (arXiv)

Machine learning is increasingly deployed in safety-critical domains where erroneous predictions may lead to potentially catastrophic consequences, highlighting the need for learning systems to be aware of how confident they are in their own predictions: in other words, 'to know when they do not know'. In this paper, we propose a novel Random-Set Neural Network (RS-NN) approach to classification which predicts belief functions (rather than classical probability vectors) over the class list using the mathematics of random sets, i.e., distributions over the collection of sets of classes. RS-NN encodes the 'epistemic' uncertainty induced by training sets that are insufficiently representative or limited in size via the size of the convex set of probability vectors associated with a predicted belief function. Our approach outperforms state-of-the-art Bayesian and Ensemble methods in terms of accuracy, uncertainty estimation and out-of-distribution (OoD) detection on multiple benchmarks (CIFAR-10 vs SVHN/Intel-Image, MNIST vs FMNIST/KMNIST, ImageNet vs ImageNet-O). RS-NN also scales up effectively to large-scale architectures (e.g. WideResNet-28-10, VGG16, Inception V3, EfficientNetB2 and ViT-Base-16), exhibits remarkable robustness to adversarial attacks and can provide statistical guarantees in a conformal learning setting.