# Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs

arXiv: https://arxiv.org/abs/2509.13813 · HTML: https://mlsanity.com/paper/2509.13813/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 45 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering](https://mlsanity.com/paper/2209.09513/) (2022)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [A Survey on LLM-as-a-Judge](https://mlsanity.com/paper/2411.15594/) (2024)
- [Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models](https://mlsanity.com/paper/2305.19187/) (2023)
- [Semantic Entropy Probes: Robust and Cheap Hallucination Detection in LLMs](https://mlsanity.com/paper/2406.15927/) (2024)
- [HuatuoGPT, towards Taming Language Model to Be a Doctor](https://mlsanity.com/paper/2305.15075/) (2023)
- [Know Your Limits: A Survey of Abstention in Large Language Models](https://mlsanity.com/paper/2407.18418/) (2024)
- [Confidence Improves Self-Consistency in LLMs](https://mlsanity.com/paper/2502.06233/) (2025)
- [Uncertainty Quantification and Confidence Calibration in Large Language Models: A Survey](https://mlsanity.com/paper/2503.15850/) (2025)
- [Uncertainty Estimation and Quantification for LLMs: A Simple Supervised Approach](https://mlsanity.com/paper/2404.15993/) (2024)
- [A Survey on Large Language Models for Critical Societal Domains: Finance, Healthcare, and Law](https://mlsanity.com/paper/2405.01769/) (2024)
- [Reasoning Aware Self-Consistency: Leveraging Reasoning Paths for Efficient LLM Sampling](https://mlsanity.com/paper/2408.17017/) (2024)
- [A Survey of Uncertainty Estimation Methods on Large Language Models](https://mlsanity.com/paper/2503.00172/) (2025)
- [CLAMBER: A Benchmark of Identifying and Clarifying Ambiguous Information Needs in Large Language Models](https://mlsanity.com/paper/2405.12063/) (2024)
- [Detecting Hallucinations in Large Language Model Generation: A Token Probability Approach](https://mlsanity.com/paper/2405.19648/) (2024)
- [K-QA: A Real-World Medical Q&A Benchmark](https://mlsanity.com/paper/2401.14493/) (2024)
- [TokUR: Token-Level Uncertainty Estimation for Large Language Model Reasoning](https://mlsanity.com/paper/2505.11737/) (2025)
- [Revisiting Uncertainty Quantification Evaluation in Language Models: Spurious Interactions with Response Length Bias Results](https://mlsanity.com/paper/2504.13677/) (2025)
- [Semantic Volume: Quantifying and Detecting both External and Internal Uncertainty in LLMs](https://mlsanity.com/paper/2502.21239/) (2025)
- [Fast and Robust Archetypal Analysis for Representation Learning](https://mlsanity.com/paper/1405.6472/) (2014)
- [Uncertainty Quantification in Large Language Models Through Convex Hull Analysis](https://mlsanity.com/paper/2406.19712/) (2024)
- [Improving Medical Diagnostics with Vision-Language Models: Convex Hull-Based Uncertainty Analysis](https://mlsanity.com/paper/2412.00056/) (2024)

## Built on by (most cited first, 13 of 13)

- [Entropy Alone is Insufficient for Safe Selective Prediction in LLMs](https://mlsanity.com/paper/2603.21172/) (2026)
- [BAS: A Decision-Theoretic Approach to Evaluating Large Language Model Confidence](https://mlsanity.com/paper/2604.03216/) (2026)
- [Semantic Self-Distillation for Language Model Uncertainty](https://mlsanity.com/paper/2602.04577/) (2026)
- [Student Guides Teacher: Weak-to-Strong Inference via Spectral Orthogonal Exploration](https://mlsanity.com/paper/2601.06160/) (2026)
- [SELFDOUBT: Uncertainty Quantification for Reasoning LLMs via the Hedge-to-Verify Ratio](https://mlsanity.com/paper/2604.06389/) (2026)
- [Uncertainty Propagation in LLM-Based Systems](https://mlsanity.com/paper/2604.23505/) (2026)
- [Beyond Majority Voting: Efficient Best-Of-N with Radial Consensus Score](https://mlsanity.com/paper/2604.12196/) (2026)
- [Dep-LLM: Training-Free Depression Diagnosis via Evidence-Guided Structured Multi-factor with Reliable LLM Reasoning](https://mlsanity.com/paper/2606.10796/) (2026)
- [Which Way from B to A: The role of embedding geometry in image interpolation for Stable Diffusion](https://mlsanity.com/paper/2511.12757/) (2025)
- [Distance Is All You Need: Radial Dispersion for Uncertainty Estimation in Large Language Models](https://mlsanity.com/paper/2512.04351/) (2025)
- [A Geometric Taxonomy of Hallucinations in LLMs](https://mlsanity.com/paper/2602.13224/) (2026)
- [Text Corpora as Concept Fields: Black-Box Hallucination and Novelty Measurement](https://mlsanity.com/paper/2605.05103/) (2026)
- [Evaluation Pitfalls and Sparsity Limitations in LLM-based Confidence Estimates for Classification](https://mlsanity.com/paper/2608.04899/) (2026)

## Abstract (arXiv)

Large language models are known to hallucinate, generating linguistically plausible but incorrect answers to questions. Uncertainty quantification has been proposed as a strategy to detect such behaviour, but existing methods lack a unified framework to assess reliability at both the prompt and answer level. We introduce a geometric framework which quantifies language model uncertainty at both levels by explicitly modelling a prompt-conditioned semantic distribution in answer embedding space. Our approach is black-box and sampling-based; we generate multiple answers per prompt, and use archetypal analysis to estimate a geometric support for the answer distribution. At the prompt level, we approximate the distribution entropy to quantify uncertainty; for each individual answer, we then use notions of atypicality to assess its reliability relative to the batch. We employ our framework to not only detect hallucinations but correct them, by selecting the batch example deemed most reliable. Experiments show that our framework performs comparably to or better than prior methods on short form question-answering datasets, and achieves superior results on medical datasets where hallucinations carry particularly critical risks. Beyond pure performance, we suggest the theoretical grounding of our work provides support for semantic distributions as useful objects of study for language model uncertainty.