# Multidimensional Uncertainty Quantification via Optimal Transport

arXiv: https://arxiv.org/abs/2509.22380 · HTML: https://mlsanity.com/paper/2509.22380/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 58 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 30 of 41)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Don't Give Me the Details, Just the Summary! Topic-Aware Convolutional Neural Networks for Extreme Summarization](https://mlsanity.com/paper/1808.08745/) (2018)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [CoQA: A Conversational Question Answering Challenge](https://mlsanity.com/paper/1808.07042/) (2018)
- [Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models](https://mlsanity.com/paper/2305.19187/) (2023)
- [AlignScore: Evaluating Factual Consistency with a Unified Alignment Function](https://mlsanity.com/paper/2305.16739/) (2023)
- [Unsupervised Quality Estimation for Neural Machine Translation](https://mlsanity.com/paper/2005.10608/) (2020)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Shifting Attention to Relevance: Towards the Predictive Uncertainty Quantification of Free-Form Large Language Models](https://mlsanity.com/paper/2307.01379/) (2023)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [DEUP: Direct Epistemic Uncertainty Prediction](https://mlsanity.com/paper/2102.08501/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Nonparametric Uncertainty Quantification for Single Deterministic Neural Network](https://mlsanity.com/paper/2202.03101/) (2022)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Uncertainty Quantification for LLMs through Minimum Bayes Risk: Bridging Confidence and Consistency](https://mlsanity.com/paper/2502.04964/) (2025)
- [Introducing an Improved Information-Theoretic Measure of Predictive Uncertainty](https://mlsanity.com/paper/2311.08309/) (2023)

## Built on by (most cited first, 1 of 1)

- [Optimal Transport Dropout for Structured Predictive Uncertainty](https://mlsanity.com/paper/2609.33377/) (2026)

## Abstract (arXiv)

Most uncertainty quantification (UQ) approaches provide a single scalar value as a measure of model reliability. However, different uncertainty measures could provide complementary information on the prediction confidence. Even measures targeting the same type of uncertainty (e.g., ensemble-based and density-based measures of epistemic uncertainty) may capture different failure modes. We take a multidimensional view on UQ by stacking complementary UQ measures into a vector. Such vectors are assigned with Monge-Kantorovich ranks produced by an optimal-transport-based ordering method. The prediction is then deemed more uncertain than the other if it has a higher rank. The resulting VecUQ-OT algorithm uses entropy-regularized optimal transport. The transport map is learned on vectors of scores from in-distribution data and, by design, applies to unseen inputs, including out-of-distribution cases, without retraining. Our framework supports flexible non-additive uncertainty fusion (including aleatoric and epistemic components). It yields a robust ordering for downstream tasks such as selective prediction, misclassification detection, out-of-distribution detection, and selective generation. Across synthetic, image, and text data, VecUQ-OT shows high efficiency even when individual measures fail. The code for the method is available at: https://github.com/stat-ml/multidimensional_uncertainty.