Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs

The evolution of LLMs like GPT-4 and PaLM has dramatically advanced NLP capabilities, enabling applications in healthcare, legal analysis, and automated reasoning. Early uncertainty quantification methods, such as self-assessment and entropy-based metrics, provided some detection ability but lacked interpretability and robustness. Recent geometric approaches, including convex hull and graph-based methods, attempted to quantify response diversity, yet often relied on linear projections or discrete heuristics, limiting their effectiveness in capturing semantic boundaries. The need for a unified, interpretable, and scalable uncertainty measure remains pressing, especially for high-stakes deployment where hallucinations can cause severe harm. This work builds on archetypal analysis and convex geometry to address these gaps, aiming to provide a comprehensive framework that is both theoretically sound and practically applicable.

Despite progress, existing uncertainty quantification techniques struggle to accurately detect hallucinations in black-box settings, especially when responses exhibit complex semantic variations. The challenge lies in capturing the true semantic boundaries of responses without internal model access, and in providing both global and local reliability estimates. Current metrics often lack interpretability, are sensitive to dimensionality reduction artifacts, or rely on heuristic thresholds. Consequently, they fail to reliably identify and correct hallucinated outputs, limiting the safe deployment of LLMs in critical domains. Developing a method that unifies global and local uncertainty assessment, grounded in geometric principles, is essential to overcome these limitations.

This work introduces a geometric framework based on archetypal analysis, which extracts extremal response prototypes (archetypes) to define the semantic boundary of response sets. The key innovations include: 1) The Geometric Volume metric, measuring the convex hull volume of archetypes as a global uncertainty indicator; 2) The Geometric Suspicion score, derived from response-archetype relationships, for local reliability assessment; 3) Theoretical proof linking convex hull volume to differential entropy, providing a rigorous foundation. Unlike prior methods, this approach offers continuous semantic boundary points, interpretable anchors, and model-agnostic operation solely based on response sampling, making it highly suitable for black-box scenarios.

• �� Generate multiple responses for a prompt using sampling at non-zero temperature. • Embed responses into a high-dimensional space with a pre-trained sentence encoder. • Apply PCA to reduce embedding dimensionality, preserving essential semantic information. • Use Archetypal Analysis to identify extremal archetypes on the convex hull of embedded responses. • Compute the convex hull volume of archetypes; take the logarithm as the global uncertainty score. • For local uncertainty, calculate response-specific metrics: distance from response to batch centroid, local density via k-nearest neighbors, and archetype reliance coefficients. • Combine these metrics using Fisher’s method to produce a suspicion score for each response. • Use a threshold on global score for hallucination detection and select the response with minimal suspicion score in Best-of-N strategies.

Experiments span datasets like MedicalQA, TriviaQA, and ScienceQA, involving multiple models (GPT-4, GPT-3.5, Llama). Metrics include F1-score, AUROC, and AUARC for response ranking. Thresholds are tuned on validation sets. Ablation studies analyze the contribution of each geometric metric. Results show that the proposed geometric volume surpasses baseline uncertainty metrics, especially in medical QA, with over 15% accuracy improvement. Local suspicion scores effectively reduce hallucinations in response selection, achieving over 20% error reduction. Theoretical validation confirms the link between convex hull volume and entropy, strengthening the method’s foundation.

The geometric volume metric consistently outperforms entropy and graph-based measures, achieving the highest F1 and AUROC scores across datasets. In high-stakes medical QA, it improves hallucination detection accuracy by over 15%. The local suspicion-based Best-of-N approach reduces hallucination rates by more than 20%, demonstrating practical utility. Theoretical analysis establishes that convex hull volume bounds the differential entropy of responses, providing a rigorous underpinning for the dispersion measure. These results collectively showcase the method’s robustness, interpretability, and scalability in diverse scenarios.

This framework can be integrated into LLM deployment pipelines in healthcare, legal, and financial sectors to flag unreliable outputs. It requires only response sampling and embedding, making it suitable for black-box models. The geometric metrics guide response filtering and selection, reducing hallucinations and increasing trustworthiness. Long-term, the approach can be extended to multi-modal data, real-time systems, and adaptive thresholding, fostering safer and more transparent AI systems in critical applications.

Computational complexity of convex hull volume calculation grows with response set size, limiting scalability. PCA reduction may discard subtle semantic cues, affecting accuracy. In highly ambiguous tasks, defining clear semantic boundaries remains challenging. Future work should optimize algorithms, incorporate multi-modal data, and develop adaptive thresholds to enhance robustness and efficiency.