Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs

## Abstract

Abstract Large language models demonstrate impressive results across diverse tasks but are still known to hallucinate, generating linguistically plausible but incorrect answers to questions. Uncertainty quantification has been proposed as a strategy for hallucination detection, but no existing black-box approach provides estimates for both global and local uncertainty. The former attributes uncertainty to a batch of responses, while the latter attributes uncertainty to individual responses. Current local methods typically rely on white-box access to internal model states, whilst black-box methods only provide global uncertainty estimates. We introduce a geometric framework to address this, based on archetypal analysis of batches of responses sampled with only black-box model access. At the global level, we propose Geometric Volume, which measures the convex hull volume of archetypes derived from response embeddings. At the local level, we propose Geometric Suspicion, which ranks responses by reliability and enables hallucination reduction through preferential response selection. Unlike prior dispersion methods which yield only a single global score, our approach provides semantic boundary points which have utility for attributing reliability to individual responses. 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. We also provide theoretical justification by proving a link between convex hull volume and entropy.

## 1 Introduction

Large language models (LLMs) have achieved remarkable performance across diverse natural language processing tasks (Guo et al., 2025; Anthropic, 2025; Gemini Team, Google DeepMind, 2025; OpenAI, 2025) and are increasingly applied in areas such as medical diagnosis, law, and financial advice (Yang et al., 2025; Chen et al., 2024; Kong et al., 2024). Hallucinations, however, where models generate plausible but false or fabricated content, pose significant risks for adoption in high-stakes applications (Farquhar et al., 2024). Recent work, for example, finds GPT-4 hallucinating in 28.6% of reference generation tasks (Chelli et al., 2024).

Uncertainty quantification (UQ) methods have been proposed to detect when models are producing unreliable outputs (Liu et al., 2025; Xiong et al., 2024). Effective UQ can serve as a critical layer of security and transparency, enabling systems to flag problematic responses and helping users exercise caution when reliability is compromised (Farquhar et al., 2024). This capability is particularly crucial for applications such as healthcare and legal services where incorrect information can cause significant harm (Huang et al., 2025; Asgari et al., 2025; Latif, 2025).

UQ methods can generally be divided into two groups: sampling-based and hidden-state-based. Sampling-based methods generate multiple responses for a prompt and estimate uncertainty over the resulting batch, requiring only black-box access. Hidden-state-based methods use intermediate activations to estimate uncertainty at the single-response level, which requires white-box access. Previous work also distinguishes between external and internal uncertainty, the former arising from ambiguous queries and the latter from insufficient model knowledge (Li et al., 2025).

We additionally distinguish between uncertainty estimates at the batch level, which we term global, and those at the individual response level, which we term local. Prior work has referred to the latter as ‘confidence’ (Lin et al., 2023). A comprehensive UQ method should provide estimates of both types. In high-stakes settings, local UQ enables the selection of the most reliable response from several alternatives, thereby reducing the risk of harmful hallucinations. We argue that archetypal analysis uniquely enables this unification. The extremal archetypes define the global semantic boundaries and also serve as anchors for local response-level diagnostics, something no prior convex-hull or entropy-based method provides.

We introduce a geometric framework to quantify global and local uncertainty using only black-box access. Our approach is sampling-based: for a given prompt we generate a batch of responses at non-zero temperature and embed them with a sentence encoder. We then apply Archetypal Analysis (AA) (Cutler & Breiman, 1994) to identify a set of archetypes that span the embedding space. Our first contribution is *Geometric Volume*, a global uncertainty metric defined as the convex hull volume of the archetypes, which reflects the semantic spread of the batch.

Our second contribution is *Geometric Suspicion*, a local uncertainty measure that accounts for the distribution of responses within the spanned volume and how each response is reconstructed from the archetypes. To our knowledge, this is the first sampling-based black-box method that differentiates high and low uncertainty responses within a batch.

We validate our framework on CLAMBER, TriviaQA, ScienceQA, MedicalQA, and K-QA. We also provide theoretical analysis linking convex hull volume to entropy. Our contributions are the following:

- •
We propose *Geometric Volume*, a global uncertainty metric that outperforms prior methods on medical QA and is competitive on standard benchmarks.
- •
We introduce *Geometric Suspicion*, the first black-box, sampling-based local uncertainty method, which reduces hallucinations by guiding Best-of-N response selection.
- •
We provide theoretical support by proving a link between the magnitude of the convex hull volume and the entropy of probability distributions defined within its boundaries.

Figure: Figure 1: A schematic of geometric volume: (1) sample $n$ responses from the LLM, (2) embed and apply dimensionality reduction, (3) perform archetypal analysis and compute the convex hull, and (4) apply a threshold to detect hallucination.
Refer to caption: x1.png

## 2 Related Works

#### Semantic Volume:

Our work is closely related to Semantic Volume (Li et al., 2025), which also analyzes the semantic dispersion of a batch of natural language outputs. Specifically, Semantic Volume computes the determinant of the Gram matrix formed from a batch of embeddings, where a low value ($\log\det(V^{\top}V)$) corresponds to low uncertainty and vice versa.

This value measures the volume of the parallelepiped formed
by the embedding vectors. We suggest that our archetype-based method leads to a more accurate representation of the embedding space spanned by the embedded responses, whilst offering useful intermediate representations (i.e. the archetypes) for calculating local uncertainty.

#### Convex Hull Approaches:

Several recent works use convex hull area over response embeddings as a proxy for uncertainty, projecting into two dimensions and summing hulls around clusters (Catak & Kuzlu, 2024; Catak et al., 2024). These works however vastly simplify the semantic space by projecting into only two dimensions with principal component analysis. They also first cluster responses before computing and summing the convex hull area around each cluster, which ignores how far apart in embedding space separate clusters may be. In doing so crucial information is lost regarding the variation in response meaning a model is likely to give to a certain prompt.

#### Semantic Entropy and Self-Consistency:

Semantic entropy detects hallucinations by clustering semantically equivalent responses using bidirectional entailment, then computing entropy over those clusters (Farquhar et al., 2024). Self-consistency methods have built on their work and are emerging as a dominant paradigm (Taubenfeld et al., 2025; Wan et al., 2025; Savage et al., 2024), where multiple responses are sampled and their agreement is used as an uncertainty signal. Again, neither of these works offer additional methods for local UQ.

#### White-box Uncertainty:

Finally, numerous methods have been proposed to estimate uncertainty with white-box model access, using for instance token probabilities, logits or hidden layer activations without requiring additional sampling. These methods include minimum token probability, average token probability, and more advanced techniques that account for the semantic importance of individual tokens (Xia et al., 2025; Zhang et al., 2025; Liu et al., 2024; Malinin & Gales, 2020; Quevedo et al., 2024).

To our knowledge, we are the first to introduce a unified framework that connects global dispersion with local attribution. Unlike prior methods that collapse geometry into a single global score, our use of archetypal analysis yields interpretable anchor points that both define batch-level uncertainty and enable principled response-level attribution in a black-box setting. This enables fine-grained hallucination detection and improves interpretability without requiring access to internal model states.

## 3 Methodology

Our geometric framework quantifies LLM uncertainty through a two-tiered geometric analysis of response embeddings. First, it computes a global uncertainty score for a set of responses by modeling the geometric boundaries of their embeddings using archetypal analysis. Second, it introduces a local uncertainty score that ranks individual responses within a sampled batch, enabling a Best-of-N strategy that replaces hallucinated zero-temperature outputs with more reliable alternatives. This is achieved without access to internal model states, making it applicable to any black-box LLM.

### 3.1 Preliminaries and Notation

Let $q$ be an input query to an LLM. We first generate a default response, $r_{\text{default}}$, using greedy decoding. We then generate a set of $n$ responses $\{r_{1},r_{2},\dots,r_{n}\}$ using a sampling strategy (e.g., non-zero temperature decoding). A pre-trained sentence embedding model (gte-Qwen2-1.5B-instruct) $\mathcal{E}:\mathcal{S}\to\mathbb{R}^{d}$ maps each response $r_{i}$ from the space of sentences $\mathcal{S}$ to a $d$-dimensional embedding vector $\mathbf{x}_{i}=\mathcal{E}(r_{i})$. To address computational challenges associated with high-dimensional embeddings, we apply L2-normalization to each $\mathbf{x}_{i}$ and then use Principal Component Analysis (PCA) to project the embeddings into a lower-dimensional subspace of dimension $d^{\prime}$. The full set of responses is represented by the data matrix $\mathbf{X}\in\mathbb{R}^{n\times d^{\prime}}$, where the $i$-th row is $\mathbf{x}_{i}^{\top}$.

### 3.2 Global Uncertainty via Geometric Volume

At a high level, we use the set of responses $\mathbf{X}$ to evaluate the uncertainty associated with $r_{\text{default}}$. Our global uncertainty metric is based on archetypal analysis (AA) (Cutler & Breiman, 1994), an unsupervised method for finding representative *archetypes* that lie on the convex hull of the data. Unlike clustering methods that identify centroids within the data cloud, archetypes represent extremal points, thereby capturing the semantic boundaries of possible model responses.

#### Archetypal Analysis Formulation

Given a data matrix $\mathbf{X}\in\mathbb{R}^{n\times d^{\prime}}$, the goal of Archetypal Analysis (AA) is to learn a set of $K$ archetypes that lie on the convex hull of the data. These archetypes form a dictionary $\mathbf{D}\in\mathbb{R}^{K\times d^{\prime}}$, with each row representing an archetype. Each data point is required to be a convex combination of the archetypes, and at the same time, each archetype must itself be representable as a convex combination of the data points.

This bi-convex constraint leads to the following optimization problem (Abrol & Sharma, 2020):

$$ $\displaystyle\operatorname*{arg\,min}_{\begin{subarray}{c}\mathbf{B},\mathbf{A}\\ \mathbf{b}_{j}\in\Delta_{n},\mathbf{a}_{i}\in\Delta_{K}\end{subarray}}\|\mathbf{X}\mathbf{-XBA}\|_{F}^{2},$ (1) $\displaystyle\Delta_{n}\triangleq[\mathbf{b}_{j}\succeq 0,$ $\displaystyle\|\mathbf{b}_{j}\|_{1}=1],\Delta_{K}\triangleq[\mathbf{a}_{i}\succeq 0,\|\mathbf{a}_{i}\|_{1}=1].$ $$

Here, $\mathbf{a}_{i}$ and $\mathbf{b}_{j}$ are columns of $\mathbf{A}\in\mathbb{R}^{K\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times K}$, respectively. Thus, the archetype dictionary is given by $\mathbf{D}=\mathbf{B}^{\top}\mathbf{X}$.

In practice, Eq. [1](https://arxiv.org/html/2509.13813v1#S3.E1) is solved iteratively via block-coordinate descent (Chen et al., 2014), alternating between updates of $\mathbf{A}$ and $\mathbf{B}$ until convergence. This procedure ensures that the resulting archetypes capture extremal directions of the data, approximating the vertices of the convex hull.

#### Geometric Volume

The learned archetypes $\mathbf{Z}=\{\mathbf{z}_{1},\dots,\mathbf{z}_{K}\}$ lie on the boundary of the space spanned by the semantic embeddings of the response set. A high degree of uncertainty, or likely hallucination, manifests as a diverse set of responses, which geometrically corresponds to archetypes that are far apart in the embedding space. We capture this dispersion by computing the volume of the convex hull of the archetypes, $V=\mathrm{volume}(\mathrm{conv}(\mathbf{Z}))$. Our global uncertainty metric, termed Geometric Volume, is the logarithm of this volume:

$$ $H_{G}(\mathbf{X})=\log\big{(}V+\epsilon\big{)}$ (2) $$

where $\epsilon$ is a small constant (e.g., $10^{-12}$) for numerical stability. A larger volume indicates greater semantic dispersion among the sampled responses, which yields a higher Geometric Volume score and signals higher uncertainty.

### 3.3 Local Uncertainty Detection

This global score $H_{G}(\mathbf{X})$ allows us to classify hallucinations successfully, but only facilitates deferral as a method of reducing hallucinations. This is because while the global score indicates if a response set is uncertain, it does not attribute uncertainty measures to individual responses within the set. Sensible measures of local uncertainty could allow reduction in hallucination rates by preferential selection of more certain responses, i.e. those less likely to be hallucinations, from the response set. To address this we introduce a simple local uncertainty measure, which we term Geometric Suspicion, composed of three separate terms derived from the archetypal coefficient matrix $\mathbf{A}$ and the answer embeddings $\mathbf{X}$.

To derive and support our choice of terms, we conduct an analysis of the datasets generated for our global uncertainty experiments, which can be found in Appendix Section [D](https://arxiv.org/html/2509.13813v1#A4).

Figure: Figure 2: We demonstrate local uncertainty metrics using Archetype Rarity, Average Distance to Nearest Neighbors, and Distance from Consensus, which can transform hallucinated responses at temperature 0 into correct responses.
Refer to caption: figures/ICLR_Figure_2_redacted.png

#### Local Density

This metric operates directly on the set of response embeddings $\mathbf{X}$. For each embedding $\mathbf{x}_{i}\in\mathbf{X}$, we identify its $k$-nearest neighbours. The score is the average Euclidean distance to these neighbours:

$$ $L(r_{i})=\frac{1}{k}\sum_{\mathbf{x}_{j}\in\mathcal{N}_{k}(\mathbf{x}_{i})}\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}.$ (3) $$

Here, $\mathcal{N}_{k}(\mathbf{x}_{i})$ is the set of the $k$-nearest neighbours to embedding $\mathbf{x}_{i}$. The intuition is that correct, coherent responses will form dense clusters (resulting in a low distance score), while hallucinations or semantic outliers will be isolated in sparse regions, yielding a high score.

#### Distance from Consensus

While Local Density is effective at finding isolated points, it can fail if a model produces a coherent but incorrect cluster of responses. To address this, our second metric establishes a global geometric context. It measures how far a response embedding $\mathbf{x}_{i}$ is from the *semantic consensus point* $\mathbf{x}_{c}$, defined as the average embedding across the response set:

$$ $D(r_{i})=\|\mathbf{x}_{i}-\mathbf{x}_{c}\|_{2},\quad\text{where}\quad\mathbf{x}_{c}=\frac{1}{n}\sum_{j=1}^{n}\mathbf{x}_{j}.$ (4) $$

This score identifies responses that are semantically distant from the central theme of the entire answer set. A high score signifies that a response, even if part of a dense local cluster, belongs to a region that is globally anomalous.

#### Usage Rarity

The final metric leverages Archetypal Analysis to identify responses that rely on fringe boundaries of the semantic space. The $i$-th row of $\mathbf{A}$, denoted $\bm{\alpha}_{i}=[A_{i1},\dots,A_{iK}]$ contains coefficients describing how the response embedding $\mathbf{x}_{i}$ is reconstructed from the set of learned archetypes $\mathbf{Z}$.
The score for a response $r_{i}$ is high if its reconstruction coefficients $\mathbf{A}_{ik}$ depend heavily on archetypes that are, on average, rarely used to explain the rest of the data.:

$$ $U(r_{i})=\sum_{k=1}^{K}A_{ik}\big{(}1-\bar{\alpha}_{k}\big{)},\quad\text{where}\quad\bar{\alpha}_{k}=\frac{1}{n}\sum_{j=1}^{n}A_{jk}.$ (5) $$

#### Final Suspicion Score

The final ‘suspicion’ score, $S(r_{i})$, is created by combining the evidence from the three individual terms in a non-parametric fashion. For each of the three metrics, we rank all responses in the batch from least suspicious (Rank 1) to most suspicious (Rank $n$). The final suspicion score for a response is the sum of its ranks across the three metrics:

$$ $S(r_{i})=\text{rank}_{L}(r_{i})+\text{rank}_{D}(r_{i})+\text{rank}_{U}(r_{i}),$ (6) $$

where $\text{rank}_{M}(r_{i})$ is the rank of response $r_{i}$ according to metric $M$ within the batch. By summing the ranks, we create a robust score that is insensitive to the individual scales of the metrics and identifies responses that are consistently flagged as anomalous. We then select the response with the minimum combined rank as the most plausible answer.

### 3.4 Hallucination Detection

Our geometric volume metric can be used to detect hallucinations at the response set level. Given a set of responses with embedding matrix $\mathbf{X}$, we classify the entire set as hallucinated if the global uncertainty exceeds a threshold $\tau$, i.e., if $H_{G}(\mathbf{X})>\tau$. The threshold $\tau$ is a tunable hyperparameter, which can be optimized on a small labeled validation set to maximize a classification objective such as F1-score.

### 3.5 Reducing Hallucination Rates with Local Uncertainty

Beyond merely identifying uncertain responses, the primary application of our suspicion score $S(r)$ is to actively improve model reliability. We employ it in a Best-of-$n$ (BoN) sampling framework, where the goal is to select the most plausible response from a set of $n$ candidates generated for a given question.

Given a set of $n$ sampled responses $R=\{r_{1},...,r_{n}\}$ for a single prompt, our uncertainty-guided selection process chooses the single response, $r^{*}$, that exhibits the minimum suspicion score. This approach replaces the model’s default (and potentially hallucinated) answer with one deemed most reliable by our proposed metric.

To quantify the efficacy of this selection strategy, we measure the net reduction in the hallucination rate across a dataset. First, we define the baseline hallucination rate, $H_{\text{baseline}}$, as the proportion of questions in a test set for which the model’s default answer $r_{\text{default}}$ was a hallucination.
Next, we define the post-selection hallucination rate, $H_{\text{BoN-U}}$, as the proportion of questions where the answer $r^{*}$ selected by our uncertainty-guided method is a hallucination.

The overall improvement is captured by the Delta Hallucination Rate, $\Delta H$, which is the absolute reduction in the hallucination rate achieved by our method.

$$ $\Delta H=H_{\text{baseline}}-H_{\text{BoN-U}}.$ (7) $$

A positive $\Delta H$ indicates that the suspicion score is successfully guiding the selection towards non-hallucinated responses, thereby demonstrating the practical utility of the local uncertainty measure.

## 4 Experiments

We conduct a series of experiments to evaluate the effectiveness of our geometric framework for hallucination detection. We assess its performance under external uncertainty (prompt ambiguity) and internal uncertainty (fact-checking). We also evaluate uncertainty methods in more realistic, long-form medical question-answering scenarios that better reflect real-world conditions.

### 4.1 Benchmarks and Baselines

We evaluate on five benchmarks covering both external and internal uncertainty:
CLAMBER (ambiguous prompts), TriviaQA (short-form QA), ScienceQA (scientific reasoning),
MedicalQA (high-stakes medical QA), and K-QA (real-world medical scenarios).
These benchmarks span ambiguous queries, factual QA, and long-form medical domains.
Dataset construction details are provided in Section [C](https://arxiv.org/html/2509.13813v1#A3) of the Appendix. We compare our method with Semantic Volume (Li et al., 2025), Semantic Entropy (Farquhar et al., 2024), and P(true), which estimates the probability that a model’s generated response is correct by directly asking the model itself to self-assess its confidence (Lin et al., 2023).

**Table 1: Results across five benchmarks and four LLMs comparing external uncertainty estimation methods. Reported values are meanstd. Blue cells indicate best results from Ours, red cells indicate best results from baselines. AUC = AUROC.**
|  | CLAMBER | TriviaQA | ScienceQA | MedicalQA | K-QA |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Method | F1 | AUC | F1 | AUC | F1 | AUC | F1 | AUC | F1 | AUC |
| GPT-4o Mini |  |  |  |  |  |  |  |  |  |  |
| $p(\mathrm{true})$ | 47.01.6 | 58.50.4 | 59.23.6 | 65.84.0 | 53.04.0 | 77.63.4 | 73.61.2 | 59.61.0 | 61.54.7 | 63.41.5 |
| Sem. Entropy | 63.90.3 | 48.10.8 | 67.73.4 | 75.30.3 | 48.62.7 | 65.31.6 | 70.91.0 | 59.31.0 | 60.44.8 | 50.55.7 |
| Sem. Volume | 68.50.4 | 55.80.9 | 69.91.0 | 74.40.3 | 54.04.4 | 57.91.5 | 73.01.2 | 59.41.0 | 68.21.6 | 65.21.1 |
| Ours | 66.10.3 | 61.70.7 | 71.11.4 | 76.70.3 | 58.73.0 | 68.82.2 | 73.61.2 | 60.91.4 | 68.60.8 | 67.71.3 |
| GPT-3.5-Turbo |  |  |  |  |  |  |  |  |  |  |
| $p(\mathrm{true})$ | 53.00.5 | 60.10.6 | 49.61.0 | 68.22.7 | 57.05.6 | 76.11.5 | 60.011.1 | 71.91.9 | 71.81.4 | 60.51.2 |
| Sem. Entropy | 66.60.2 | 53.80.2 | 69.61.2 | 75.60.4 | 62.33.3 | 69.21.5 | 71.31.5 | 68.70.3 | 40.36.9 | 66.41.2 |
| Sem. Volume | 68.50.1 | 54.40.6 | 71.70.6 | 74.90.8 | 64.33.2 | 62.74.5 | 73.10.3 | 61.70.6 | 74.81.0 | 65.30.7 |
| Ours | 82.52.6 | 61.21.3 | 71.81.5 | 76.70.4 | 70.13.7 | 73.43.6 | 73.91.1 | 64.90.4 | 75.40.4 | 67.83.2 |
| Qwen3-8b |  |  |  |  |  |  |  |  |  |  |
| $p(\mathrm{true})$ | 55.00.8 | 57.10.3 | 64.13.6 | 86.01.1 | 41.34.3 | 66.61.2 | 74.41.3 | 66.91.7 | 72.80.8 | 66.31.4 |
| Sem. Entropy | 63.80.2 | 49.30.5 | 78.10.6 | 84.20.3 | 22.219.3 | 58.90.8 | 73.20.5 | 71.10.6 | 73.41.1 | 72.50.8 |
| Sem. Volume | 67.10.2 | 55.30.2 | 74.41.4 | 80.80.7 | 66.60.1 | 48.10.6 | 74.40.8 | 69.10.3 | 73.31.1 | 69.80.2 |
| Ours | 65.90.1 | 59.10.4 | 76.90.3 | 82.10.4 | 66.70.2 | 67.20.4 | 75.21.1 | 69.50.6 | 73.61.2 | 69.80.2 |
| Llama3.1-8b |  |  |  |  |  |  |  |  |  |  |
| $p(\mathrm{true})$ | 62.82.0 | 62.00.5 | 60.82.7 | 58.212.2 | 45.32.9 | 69.12.7 | 74.42.2 | 76.42.1 | 55.83.8 | 67.94.0 |
| Sem. Entropy | 67.10.4 | 58.00.7 | 71.01.0 | 76.90.6 | 73.00.8 | 77.00.8 | 76.60.7 | 72.01.2 | 82.81.9 | 69.92.5 |
| Sem. Volume | 66.80.1 | 51.00.4 | 70.60.0 | 72.70.7 | 66.40.2 | 75.40.7 | 75.40.2 | 56.20.7 | 81.70.4 | 65.02.2 |
| Ours | 76.22.1 | 54.91.2 | 71.30.5 | 75.40.5 | 78.60.7 | 81.60.4 | 75.30.2 | 67.20.9 | 82.70.1 | 72.81.1 |

#### Local Uncertainty Evaluation

We evaluate our local uncertainty quantification methods on the same datasets curated for the global Internal Uncertainty experiments. We first filter each dataset to only contain questions where the set of sampled responses contains examples of both hallucinations and non-hallucinations. For each sample in the perturbed responses, we then compute the local uncertainty score using the archetypal reconstruction matrix $\mathbf{A}$ and embeddings $\mathbf{X}$.

**Table 2: Geometric Suspicion performance on (left) all valid samples ($0<H<1$) and (right) mid-hallucination samples ($0.33<H<0.67$). Values are hallucination rate (HR, %), subscripts indicate standard error. $\Delta$HR is the absolute reduction in hallucination rate after Best-of-N selection. Bold indicates the strongest performance in each dataset column.**
|  |  | All Valid ($0<H<1$) | Mid-Valid ($0.33<H<0.67$) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Dataset | Model | Baseline HR | BoN HR | $\Delta$HR | Baseline HR | BoN HR | $\Delta$HR |
| K-QA |  |  |  |  |  |  |  |
|  | GPT-3.5-Turbo | 60.41.2 | 29.15.3 | 31.34.4 | 65.08.7 | 40.310.6 | 24.74.3 |
|  | GPT-4o Mini | 48.52.2 | 33.42.2 | 15.13.4 | 54.52.5 | 36.97.4 | 17.77.0 |
|  | Llama3.1-8b | 66.21.2 | 51.51.8 | 14.73.0 | 62.40.9 | 45.94.7 | 16.55.3 |
|  | Qwen3-8b | 55.52.9 | 36.84.6 | 18.77.3 | 64.82.5 | 35.22.5 | 29.64.9 |
| MedicalQA |  |  |  |  |  |  |  |
|  | GPT-3.5-Turbo | 52.51.7 | 43.71.1 | 8.90.7 | 57.42.3 | 39.52.4 | 17.94.5 |
|  | GPT-4o Mini | 48.71.5 | 39.70.8 | 9.01.0 | 50.93.8 | 42.03.3 | 8.90.6 |
|  | Llama3.1-8b | 52.91.8 | 47.22.3 | 5.70.8 | 42.93.3 | 29.45.5 | 13.52.3 |
|  | Qwen3-8b | 52.03.2 | 44.62.1 | 7.41.4 | 50.24.3 | 37.42.9 | 12.86.0 |
| ScienceQA |  |  |  |  |  |  |  |
|  | GPT-3.5-Turbo | 80.64.7 | 77.87.0 | 2.82.5 | 77.94.8 | 68.77.9 | 9.25.2 |
|  | GPT-4o Mini | 91.24.9 | 85.05.3 | 6.21.0 | 96.53.0 | 76.65.0 | 19.92.0 |
|  | Llama3.1-8b | 85.20.3 | 77.43.3 | 7.83.2 | 85.82.6 | 65.59.7 | 20.38.5 |
|  | Qwen3-8b | 92.31.8 | 88.61.6 | 3.73.3 | 100.00.0 | 80.818.8 | 19.218.8 |
| TriviaQA |  |  |  |  |  |  |  |
|  | GPT-3.5-Turbo | 46.20.8 | 40.20.4 | 6.00.7 | 47.94.2 | 26.92.7 | 21.14.3 |
|  | GPT-4o Mini | 52.91.0 | 46.20.5 | 6.61.5 | 44.32.2 | 22.80.5 | 21.51.7 |
|  | Llama3.1-8b | 55.71.5 | 53.71.3 | 2.01.7 | 47.92.5 | 38.52.3 | 9.41.1 |
|  | Qwen3-8b | 45.00.2 | 45.60.9 | -0.50.7 | 30.10.9 | 27.41.5 | 2.71.8 |

## 5 Results

We evaluate our geometric framework for hallucination detection across multiple benchmarks and models. For global uncertainty, we test the Geometric Volume’s ability to classify response sets as reliable or hallucinated. For local uncertainty, we assess the Geometric Suspicion score in reducing hallucinations via a Best-of-N selection strategy. Each experiment is repeated three times to mitigate sampling stochasticity, and we report the mean and standard deviation of all metrics.

### 5.1 Global Uncertainty Detection

Geometric Volume shows strong performance across all benchmarks and models investigated. On K-QA, it significantly outperforms baselines, achieving the best F1 Score and AUROC with GPT-3.5-Turbo (75.4 and 67.8) and the strongest AUROC with GPT-4o Mini (67.7) and Llama3.1-8b (72.8). On MedicalQA, it remains highly competitive, delivering top F1 Scores with GPT-3.5-Turbo (73.9) and Qwen3-8b (75.2), as well as the best AUROC with GPT-4o Mini (60.9) (Table [1](https://arxiv.org/html/2509.13813v1#S4.T1)).

On CLAMBER, which tests detection of ambiguous prompts, Geometric Volume achieves the highest AUROC with GPT-4o Mini (61.7) and Qwen3-8b (59.1), while GPT-3.5-Turbo and Llama3.1-8b reach the highest F1 Scores. In TriviaQA, it results in the highest F1 and AUROC with GPT-4o Mini (71.1, 76.7) and GPT-3.5-Turbo (71.8 F1, 76.7 AUROC). On ScienceQA, it shows particular strength with Llama3.1-8b, where it delivers 78.6 F1 and 81.6 AUROC. These results highlight its ability to model semantic dispersion effectively.

### 5.2 Reducing Hallucination Rates with Local Uncertainty

We show in Table [2](https://arxiv.org/html/2509.13813v1#S4.T2) that using our local uncertainty metrics, we are able to reduce hallucination rates across all models and datasets investigated. Our method is particularly effective in ‘mid-hallucination’ subsets (described further in Section [6](https://arxiv.org/html/2509.13813v1#S6)), which would be the primary intended use case for our hallucination reduction method, deployed in conjunction with a global uncertainty method for hallucination detection.

## 6 Discussion

Our local uncertainty framework has varying efficacy depending on the context in which it is applied. Let us consider three different situations of sampling-based uncertainty analysis, where for a given input we sample multiple times with a non-zero temperature.

Firstly, a ‘high-hallucination’ case, where the majority of sampled answers are in fact hallucinations. In this case, we would expect a high semantic dispersion, resulting in a high global uncertainty score. In such a case, particularly for safety-critical applications such as healthcare, we might decide that the model has insufficient capacity to answer the question, and escalate or defer the issue at hand.
Secondly, a ‘low-hallucination’ case, where a minority of sampled answers are hallucinations. This should result in low semantic dispersion and low global uncertainty score, in which case we could propagate the original, zero-temperature response.
Finally, the ‘mid-hallucination’ case, where there is no clear majority of hallucinations or non-hallucinations, and global uncertainty analysis yields a middling score. In these cases the model may have sufficient capacity to answer the question well, but we might not be confident in the original, zero-temperature response. These are the crucial cases where we can leverage local uncertainty scores to reduce hallucination rates.

Figure: (a) A low rank for each component
Refer to caption: figures/local_uncertainty_examples/greek_coffee.png

In Figure [3](https://arxiv.org/html/2509.13813v1#S6.F3), we present t-SNE plots of cases where our local uncertainty framework was able to ‘flip’ a model response from a hallucination to a non-hallucination. The different examples show how the component terms of Geometric Suspicion contribute to selecting a desirable sample from the batch.

Figure [3](https://arxiv.org/html/2509.13813v1#S6.F3)(a) shows a straightforward example where all three of the component metrics rank highly. The correct answer is in a dense local cluster, near a global consensus point, and can be reconstructed using commonly used archetypes.
Figure [3](https://arxiv.org/html/2509.13813v1#S6.F3)(b) shows a slightly harder example, where there are multiple clusters of semantically similar answers (all are countries or some slight variation thereof). Here the desired answer is not within the densest local cluster or the closest to a global consensus, but has lower suspicion due to its non-reliance on rarely used archetypes.
The example in Figure [3](https://arxiv.org/html/2509.13813v1#S6.F3)(c) can be used to understand the necessity of both local density and distance from consensus metrics. Here we have a few dense clusters, resulting in high local density for the majority of the batch. The global semantic consensus point however is closer to the ‘Indonesia’ cluster than it is the ‘Not specified’ or ‘Unknown entity’ clusters. This is because the hallucinated answers of ‘Brazil’ or ‘Pakistan’ are countries and semantically closer to the correct answer, moving the global consensus further in this direction.
Finally, Figure [3](https://arxiv.org/html/2509.13813v1#S6.F3)(d) shows our framework can be useful in cases of diverse answer batches without dense local clusters. In this example approximately half of the answers are semantically unique, but the use of commonly used archetypes in the ‘W.H. Auden’ cluster allows us to find a low suspicion example which proves to be correct.

## 7 Conclusion

We presented a geometric framework for detecting and mitigating hallucinations in large language models using only black-box access. At the global level, Geometric Volume quantifies semantic dispersion through convex hull analysis of archetypes and provides a principled proxy for uncertainty. At the local level, Geometric Suspicion identifies unreliable responses within a batch and enables a Best-of-N selection strategy that consistently reduces hallucination rates across benchmarks. Our experiments demonstrate strong performance in both standard QA tasks and high-stakes medical domains, while our theoretical analysis establishes a direct connection between convex geometry and entropy. These results highlight geometry as a powerful tool for linking global dispersion with local reliability, and open new directions for uncertainty estimation that are interpretable, data-efficient, and broadly applicable.

## 8 Acknowledgments

DC was funded by an NIHR Research Professorship; a Royal Academy of Engineering Research Chair; and the InnoHK Hong Kong Centre for Cerebro-cardiovascular Engineering (COCHE); and was supported by the National Institute for Health Research (NIHR) Oxford Biomedical Research Centre (BRC) and the Pandemic Sciences Institute at the University of Oxford. EP was funded by an NIHR Research Studentship. SW is supported by the Rhodes Scholarship. This work was supported by the NVIDIA Academic Grant Program.

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## Appendix A Theoretical Analysis

We now present a theoretical analysis to formally justify the different components for the proposed uncertainty estimation framework.

### A.1 Simplicial Volume as a Proxy for Uncertainty

To capture the geometric structure underlying LLM responses, we represent each response as a convex combination of learned archetypes, which lie at the vertices of a simplex in the embedding space. The response set lies within the simplex spanned by these archetypes, whose volume reflects the semantic dispersion involved in its generation. Intuitively, a larger simplex volume implies more “room” for diverse responses, geometrically constraining how concentrated the underlying distribution can be. This aligns with differential entropy, which quantifies the spread or unpredictability of a continuous distribution.

We now formalize this relationship by proving that the volume of the archetypal simplex provides an upper bound on the differential entropy of any distribution supported within it, thereby linking geometric dispersion to semantic uncertainty.

###### Theorem 1 .

Let $\mathcal{A}=\{\mathbf{a}_{1},\dots,\mathbf{a}_{k}\}\subset\mathbb{R}^{d}$ be a set of $k$ affinely independent archetypes. Let $\Delta=\operatorname{conv}(\mathcal{A})$ denote the $(k{-}1)$-dimensional simplex they span, with intrinsic volume $V>0$ measured using the $(k{-}1)$-dimensional Hausdorff measure $\mathcal{H}^{k-1}$ on the affine span of $\Delta$. Let $\mathbf{x}$ be a continuous random vector supported on $\Delta$, with density $p(\mathbf{x})$ that is absolutely continuous with respect to $\mathcal{H}^{k-1}$. Then the differential entropy of $\mathbf{x}$ satisfies:

$$ $H(\mathbf{x})=-\int_{\Delta}p(\mathbf{x})\log p(\mathbf{x})\,d\mathcal{H}^{k-1}(\mathbf{x})\leq\log V,$ (8) $$

and the upper bound is achieved if and only if $\mathbf{x}$ is uniformly distributed over $\Delta$.

###### Proof.

See Appendix  [B](https://arxiv.org/html/2509.13813v1#A2).
∎

###### Corollary 1 .

Let a response be represented as a convex combination of archetypes lying in a simplex $\Delta\subset\mathbb{R}^{d}$ with volume $V>0$. Then the differential entropy of $\mathbf{x}$, with density $p(\mathbf{x})$ supported on $\Delta$, satisfies:

$$ $H(\mathbf{x})\leq\log V.$ (9) $$

Consequently, the volume $V$ provides a geometric upper bound on the differential entropy of a response distribution defined within the simplex. Differential entropy in this context is analogous to the semantic uncertainty of a response set. This bound is computationally efficient to estimate and can be used to detect low-confidence or out-of-distribution inputs without explicitly modeling $p(\mathbf{x})$.

This result follows directly from Theorem [1](https://arxiv.org/html/2509.13813v1#Thmtheorem1), by applying the entropy–volume bound to the simplex associated with the response’s archetypal representation.

###### Theorem 1 .

###### Proof.

###### Corollary 1 .

## Appendix B Proof of Theorem 1

###### Proof.

Let $u(\mathbf{x})=\frac{1}{V}$ denote the uniform distribution over $\Delta$, where $V$ is the volume of $\Delta$ under the $(k{-}1)$-dimensional Hausdorff measure $\mathcal{H}^{k-1}$. The differential entropy of $u$ is:

$$ $H(u)=-\int_{\Delta}\frac{1}{V}\log\left(\frac{1}{V}\right)d\mathcal{H}^{k-1}(\mathbf{x})=\log V,$ (10) $$

since $\int_{\Delta}d\mathcal{H}^{k-1}(\mathbf{x})=V$.

The Kullback–Leibler (KL) divergence from $p$ to $u$ is:

$$ $D_{\mathrm{KL}}(p\|u)=\int_{\Delta}p(\mathbf{x})\log\left(\frac{p(\mathbf{x})}{u(\mathbf{x})}\right)d\mathcal{H}^{k-1}(\mathbf{x}).$ (11) $$

Substituting $u(\mathbf{x})=\frac{1}{V}$ yields:

$$ $D_{\mathrm{KL}}(p\|u)=\int_{\Delta}p(\mathbf{x})\log p(\mathbf{x})\,d\mathcal{H}^{k-1}(\mathbf{x})+\log V.$ (12) $$

Rearranging terms, we obtain:

$$ $H(\mathbf{x})=-\int_{\Delta}p(\mathbf{x})\log p(\mathbf{x})\,d\mathcal{H}^{k-1}(\mathbf{x})=\log V-D_{\mathrm{KL}}(p\|u).$ (13) $$

By the non-negativity of KL divergence (Gibbs’ inequality), $D_{\mathrm{KL}}(p\|u)\geq 0$, with equality if and only if $p=u$ everywhere. Thus:

$$ $H(\mathbf{x})\leq\log V,$ (14) $$

with equality if and only if $\mathbf{x}$ is uniformly distributed over $\Delta$.
∎

###### Proof.

## Appendix C Experimental Setup

### C.1 Benchmarks and Baselines

#### CLAMBER (External Uncertainty).

To evaluate the detection of external hallucinations arising from ambiguous user prompts, we use the CLAMBER benchmark (Zhang et al., 2024). We use 3,202 queries from the dataset, where the model’s uncertainty should reflect the ambiguity of the input rather than a lack of internal knowledge.

#### TriviaQA (Internal Uncertainty).

We also evaluate on TriviaQA, a short-form question answering benchmark (Joshi et al., 2017). From the TriviaQA dataset, we construct a balanced test set of 1,000 samples, consisting of 500 hallucinated and 500 non-hallucinated examples.

#### ScienceQA Multiple Choice QA

We also used ScienceQA as a benchmark (Lu et al., 2022), which is a multimodal multiple-choice dataset designed to assess scientific reasoning across natural, social, and language sciences. For our evaluation, we filtered the test set to include only text-only questions and created a balanced benchmark of 200 hallucinating and 200 non-hallucinating examples for each LLM.

#### MedicalQA (High-Stakes Domain).

To assess performance in a critical, high-stakes domain, we use a medical question-answering benchmark subset (500 examples) from MedQA and MedMcQA (Zhang et al., 2023). This dataset tests the model’s ability to provide factually correct answers to medical questions, where hallucinations carry significant risk. MedicalQA is a long-form question answering benchmark, unlike TriviaQA, and provides deeper insight into hallucination in real-world applications.

#### K-QA (Real-World Medical Scenarios).

Finally, to evaluate uncertainty in real-world medical scenarios, we leverage 201 real patient questions from the K-QA dataset (Manes et al., 2024), a long-form generation benchmark with expert-written answers reviewed by licensed physicians. These detailed responses serve as high-quality references for assessing the accuracy and reliability of model outputs.

### C.2 Dataset Curation

#### External Uncertainty.

For the CLAMBER benchmark, we evaluate each of the 3,202 queries, where each $q_{i}$ is annotated with a binary label $y_{i}\in{0,1}$ indicating whether it is ambiguous. We generate $n=20$ perturbations $\{q_{i}^{(j)}\}_{j=1}^{n}$ for each query by prompting each LLM following previous work (Li et al., 2025), and obtain embeddings $\{x_{i}^{(j)}\}_{j=1}^{n}$ via the Alibaba-NLP/gte-Qwen2-1.5B-instruct model.

#### Internal Uncertainty.

To create hallucination detection datasets from traditional QA benchmarks, we convert question–answer pairs ${(q_{i},a_{i}^{\text{ref}})}$ into labeled examples. For each question $q_{i}$, we generate a response $r_{i}=\text{LLM}(q_{i})$ using LLM. We then assign a hallucination label $y_{i}\in\{0,1\}$ by comparing $r_{i}$ to the reference answer $a_{i}^{\text{ref}}$. In high-stakes domains such as medical QA, this comparison is performed by an LLM-as-a-judge (Gu et al., 2024) (GPT-4o) using expert-annotated references. For example, given a real-world clinical question and its verified answer, we label $r_{i}$ as hallucinated if the LLM judge determines it deviates from the reference. For datasets like TriviaQA, where answers are typically short (1–3 words), we instead compute ROUGE-L F1 and label $r_{i}$ as hallucinated if $\text{ROUGE}(r_{i},a_{i}^{\text{ref}})<0.3$, following previous work (Li et al., 2025).
When initially answering questions, we set the generation temperature to 0. Later, for sampling perturbed responses, we use a temperature of 1.0 to induce diversity. Finally, for each $q_{i}$, we sample $n=20$ perturbed responses $\{r_{i}^{(j)}\}_{j=1}^{n}$ and embed them via $\{\mathbf{x_{i}^{(j)}}=E(r_{i}^{(j)})\}_{j=1}^{n}$, using the same embedding pipeline as in the external uncertainty setting.

### C.3 Implementation Details

We set the PCA dimension to 15 and number of archetypes $K=16$ for each benchmark. The Archetypal Analysis optimization is run for 2000 steps on each batch of sampled responses. For a fair comparison, both our method and the Semantic Volume baseline require an uncertainty threshold $\tau$ to make a binary prediction (hallucination vs. not) (Li et al., 2025). Following prior work, we tune this threshold on a held-out validation split (10% of the data), selecting the $\tau$ that maximizes the F1-score. Local uncertainty experiments are performed with $k=5$ nearest neighbours for the local density metric.

## Appendix D Selection of Geometric Suspicion Terms

Our Geometric Suspicion score is composed of three separate terms. To determine a suitable choice for these terms, we conducted an analysis of our datasets generated for the global uncertainty evaluation.

For each model $M$ and dataset $D$ investigated, we have a set of $N_{D,M}$ questions. For each question, we have a ‘default’ answer, sampled with a temperature $T=0$, and a corresponding hallucination label as determined by a judge LLM (as described in Appendix Section [C](https://arxiv.org/html/2509.13813v1#A3)). For each question, we have in addition $n=20$ answers sampled with temperature $T=1$, for which we generate hallucination labels as for the default answer. For each batch of answers, we then have the objects derived from our archetypal analysis: the answer embeddings $\mathbf{X}$, the convex decomposition coefficients $\mathbf{A}$, and the archetype embeddings $\mathbf{Z}$.

The primary intended application of our local uncertainty metric is hallucination reduction, so we first look at cases where there is a possibility to ‘flip’ a model response to a non-hallucination by preferential selection of one of the batch responses. Figure [4](https://arxiv.org/html/2509.13813v1#A4.F4) plots a histogram of the sampled hallucination rate to determine the opportunity for doing so within our datasets. In the vast majority of cases, when the model hallucinates in the default answer, a high proportion of the sampled answers are also hallucinations. Approximately half the time, however, at least one answer is a non-hallucination, providing an opportunity for an informed local metric to be of use.

Figure: Figure 4: Distribution across all models and datasets of $T=1$ hallucination rate when the $T=0$ answer is hallucination.
Refer to caption: figures/hrate_dist.png

Next, we aimed to find some high level characteristics of hallucinations within our response set. We selected cases where the default response was judged to be a hallucination, but at least one hallucination and non-hallucination was present in the batch of $T=1$ answers. On the basic principle that for simple QA datasets there are many ways to be wrong, and fewer ways to be right, we formulated two hypotheses regarding the nature of hallucinations within the context of the embedding space spanned by each batch of answers. Firstly, hallucinations should sit in sparser regions of the embedding space compared to non-hallucinations. Secondly, hallucinations should sit closer to the edges of the batch embedding space, and therefore closer to the archetypes.

To test these, we compiled several terms of interest and compared them across our datasets. To measure the density of points within the embedding space, we used Local Density with $k=5$ (as described in Section [3.3](https://arxiv.org/html/2509.13813v1#S3.SS3)) and Voronoi cell volume. The Voronoi cell of an embedding $\mathbf{x}_{i}$ is the region of space closer to $\mathbf{x}_{i}$ than to any other point in the batch. A larger cell volume implies the response lies in a sparser region. As direct computation of these volumes in high-dimensional space is intractable, we approximate them via the dual Delaunay triangulation. First, for each batch of embeddings $\mathbf{X}=\{\mathbf{x}_{1},\dots,\mathbf{x}_{N}\}$, we reduce their dimensionality to a low-dimensional space (typically 3D) using Principal Component Analysis (PCA). We then compute the Delaunay triangulation of these reduced points. The volume of a response’s Voronoi cell is approximated by summing the volumes of all Delaunay simplices that have the response’s embedding as a vertex. A larger estimated volume signifies that a response lies in a sparser region of the local embedding space and therefore might be considered more suspicious.

To assess global position within the embedding space, we used the Distance from Consensus term described in Section [3.3](https://arxiv.org/html/2509.13813v1#S3.SS3) and Distance to Closest Archetype. The latter assesses a response’s proximity to the boundaries of the semantic space spanned by the batch. For each response embedding $\mathbf{x}_{i}$, we compute its Euclidean distance to every archetype embedding $\mathbf{z}_{k}$, and take the minimum. A smaller distance means the response is closer to the edges of the batch semantic space and indicates greater suspicion. To align this with our other metrics where higher values denote higher suspicion, the final score is the negative of this minimum distance, defined as:

$$ $D_{A}(r_{i})=-\min_{k\in\{1,\dots,K\}}\|\mathbf{x}_{i}-\mathbf{z_{k}}\|_{2}$ (15) $$

To assess the archetypal composition, we used Usage Rarity, as described in Section [3.3](https://arxiv.org/html/2509.13813v1#S3.SS3) and Geometric Entropy. The latter is the Shannon entropy (Shannon, 1948) of the archetypal reconstruction coefficients. The $i$-th row of $\mathbf{A}$, denoted $\bm{\alpha}_{i}=[A_{i1},\dots,A_{iK}]$, is a probability distribution describing how the pre-processed response embedding $\mathbf{x}_{i}$ is reconstructed from the set of learned archetypes $\mathbf{Z}$. A response that aligns clearly with a single archetype will have a sparse, low-entropy $\bm{\alpha}_{i}$, while a response that lies between multiple archetypes will have a more uniform, high-entropy $\bm{\alpha}_{i}$.

The Geometric Entropy for an individual response $r_{i}$ is defined as:

$$ $H_{L}(r_{i})=H(\bm{\alpha}_{i})=-\sum_{k=1}^{K}A_{ik}\log A_{ik}$ (16) $$

Additionally, we split our dataset into subsets based on the sampled hallucination rate. As behaviour such as clustering could be very different depending on whether 1 or 19 of the 20 responses were hallucinations, we split the dataset into low, mid-low, mid-high, and high hallucination subsets, with the corresponding rates described in Table [3](https://arxiv.org/html/2509.13813v1#A4.T3).

**Table 3: Definition of sampled hallucination rate subsets used for granular analysis. The rate, $r$, refers to the fraction of hallucinated responses within a 20-sample set for a given question.**
| Subset Name | Sampled Hallucination Rate ($r$) |
| --- | --- |
| Low | $0<r\leq 0.25$ |
| Mid-Low | $0.25<r\leq 0.50$ |
| Mid-High | $0.50<r\leq 0.75$ |
| High | $0.75<r<1.0$ |

For each term and subset of interest, we then compared the distribution of term values for hallucinations versus non-hallucinations in the sampled answers. In calculating each term, we constructed the sign such that our hypothesis for its predictive value aligned with our local uncertainty framework: a high term value should lead to high suspicion of being a hallucination. Finally, we performed a one-sided Mann-Whitney U Test test to determine whether the term scores for hallucinations were significantly greater than for non-hallucinations. Table [4](https://arxiv.org/html/2509.13813v1#A4.T4) shows the results.

**Table 4: One-sided p-values from the Mann-Whitney U test, testing if metric scores for hallucinations are stochastically greater than for non-hallucinations across the defined data subsets (see Table [3](https://arxiv.org/html/2509.13813v1#A4.T3)). An extremely small p-value (e.g., $<10^{-100}$) indicates a metric is both statistically significant and directionally correct. Values of $1.000$ indicate the metric is significant in the opposite direction (lower scores for hallucinations).**
|  | Sampled Hallucination Rate Subset |  |  |  |
| --- | --- | --- | --- | --- |
| Uncertainty Metric | Low | Mid-Low | Mid-High | High |
| Distance Consensus | $<10^{-100}$ | $<10^{-100}$ | $1.000$ | $1.000$ |
| Local Density | $<10^{-100}$ | $<10^{-100}$ | $0.0152$ | $1.000$ |
| Usage Rarity | $<10^{-100}$ | $<10^{-100}$ | $0.0544$ | $1.000$ |
| Voronoi Volume | $<10^{-100}$ | $<10^{-100}$ | $0.0739$ | $1.000$ |
| Geometric Entropy | $<10^{-100}$ | $<10^{-100}$ | $0.2903$ | $1.000$ |
| Distance Nearest Archetype | $<10^{-100}$ | $<10^{-100}$ | $0.2533$ | $1.000$ |

For all of our terms of interest, and hallucination rates less than 0.5, our intuitions were proved correct: non-hallucinations lie in denser regions of semantic space, are closer to a global consensus, and generally lie far away from archetypes and the edge of the embedding space. For high hallucination rates (over 0.75) this behaviour is completely reversed. We accept that in these cases it may be extremely hard to find the needle non-hallucination in a haystack of hallucinations. We suggest that in such situations it may be preferential to first use a global uncertainty measure to identify that the batch has high semantic dispersion, and defer from answering, as described in Section [6](https://arxiv.org/html/2509.13813v1#S6).

For the mid-high subset, for all our terms of interest bar ‘Distance from Consensus’ there is inconclusive evidence that hallucinations have greater values, although Local Density and Usage Rarity have p-values of 5% or below. This suggests potential to diagnose hallucinations within the batch even at these high hallucination rates, using a combination of multiple terms.

We choose Local Density and Usage Rarity as components of Geometric Suspicion due to their relatively high performance in the Mann-Whitney U tests, and their alignment with our original intuitions. We further include Distance from Consensus based on empirical observations of multiple clusters in some response batches, corresponding to both hallucinations and non-hallucinations. In these cases we find the Distance from Consensus term helps to select the non-hallucinating cluster.

## Peer Reviews

confidence: 5
contribution: 2
initial_rating: 4
presentation: 2
questions: 1.	Can you add more discussions and compare to Semantic density [1], which is also a sampling-based response-wise confidence/uncertainty metric without access to the internal LLM state? Moreover, please also consider comparing to Degree [2] and the length-normalized likelihood [3].
2.	When converting the responses into embedding space, the prompt is not considered as a context. This may cause problems for measuring the semantic relationships among the responses. One example: Q: “What is the capital of France?” A1: “Paris”, A2: “the capital of France is Paris”. A1 and A2 actually mean the same thing under this context, but they will have very different embeddings without considering the original questions as context. Have you considered this limitation? 
3.	In the "Convex Hull Approaches" part of “Related Works”, existing works are criticized to oversimplify the problem by using PCA to reduce the embedding dimensionality to two. However, the proposed approach also uses PCA to reduce the embedding dimension (to 15 dimensions). How do you make sure 15 dimensions are sufficient to preserve important information? Do you have an ablation study with different PCA dimensions?
4.	The likelihood of each sampled response is not utilized in the local uncertainty calculations. Using this information can potentially reduce the cost of sampling, i.e., we don’t need to sample the same response multiple times to estimate the output distribution. What is your consideration here?
5.	In the experimental setup, it is stated “to classify response sets as reliable or hallucinated…”. What do you exactly mean here? Are you using the default answer to represent the response set?
6.	Did you handle the long-form question/answer in medicalQA in a different way than other shorter QA benchmarks? If not, do you think one uncertainty score for the entire long answer, which may include multiple claims, is sufficient?
7.	Do you have an explanation or analysis about why the answer selection does not work well in Qwen3-8B? What makes the performance difference so different across different model?

[1] Xin Qiu, Risto Miikkulainen. Semantic density: Uncertainty quantification for large language models through confidence measurement in semantic space, Advances in Neural Information Processing Systems (NeurIPS), 2024

[2] Zhen Lin, Shubhendu Trivedi, Jimeng Sun, Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models, Transactions on Machine Learning Research, 2024

[3] Kenton Murray, David Chiang. Correcting Length Bias in Neural Machine Translation. In Proceedings of the Third Conference on Machine Translation, 2018.
soundness: 2
strengths: 1.	Using archetypal analysis to quantify the uncertainty of LLM responses provides a new perspective.
2.	Providing response-wise uncertainty is of more practical values than existing prompt-wise uncertainty metrics.
summary: This manuscript presents a new framework to quantify the uncertainty of LLM responses from a geometric perspective. The framework provides both a group-wise uncertainty score and a response-wise uncertainty score based on archetypal analysis. Experiments on several QA benchmark are performed to validate the effectiveness of the approach.
weaknesses: 1.	The main claim “this is the first sampling-based black-box method that differentiates high and low uncertainty responses within a batch.” is not true. See [1], where the proposed semantic density metric also measures sampling-based response-wise confidence/uncertainty without accessing the internal LLM state.
2.	The design of the local uncertainty, i.e., the sum of ranks in three simple heuristic metrics, is not supported by any solid mathematical justifications. 
3.	The interpretation of the experimental results need to be further clarified. See “Questions” below.
4. Several related baseline methods are missing in the current experiments. See “Questions” below for more details.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 3
questions: see above.
soundness: 2
strengths: - The theoretical framework (Appendix A)  to justify allows for a higher-level understanding of the ideas proposed in the paper. I would suggest you find a way to include this in the main paper. 
- The benchmarks used to evaluate the framework are adequate and diverse. Although most are focused on medical data, there are more general purpose datasets used as well.
summary: The authors propose a hallucination detection method that lies within a geometric framework that uses black-box model access. The framework includes both global uncertainty estimates through measuring geometric volume, while local uncertainty estimates use geometric suspicion. In doing this, the framework is able to capture both global and local uncertainty in the LLM responses, quantifying the reliability of the response through semantic boundary points.
weaknesses: - The performance, as reported in table 1, is not consistently higher than other baselines. More experiments need to be conducted to understand why this is the case. It seems that P(True), the simplest baseline out of all of them, outperforms in certain scenarios, so considering the complexity associated with the proposed approach in comparison with P(True), there needs to be better justification by the others. 
- A complexity and/or time analysis of the framework is missing from the paper.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: see weakness
soundness: 3
strengths: This paper proposes a novel and conceptually interesting perspective on hallucination detection and mitigation through geometric modeling of uncertainty. It introduces a distinction between global and local uncertainty, which offers a potentially useful framework for understanding LLM confidence. The paper also propose an appealing high-level idea to address hallucination detection and mitigation in a unified framework.
summary: This paper introduces a geometric framework for understanding uncertainty in large language models, distinguishing between global and local uncertainty. The authors propose a black-boxed method to both detect and mitigate hallucinations by modeling uncertainty from a holistic perspective. The paper evaluates the approach on several benchmarks, showing moderate performance improvements in some cases.
weaknesses: (1) My main concern is the lack of essential ablations. As a sampling based method, key hyperparameters such as sampling temperature and number of samples are not studied, but rather fixed to the same value through out the paper. This makes it difficult to assess the generalizability of the proposed approach.

(2) While the authors highlight the importance of medical data, the proposed method on MedicalQA underperforms baselines in AUROC on most models in Table 1.  Also, there's typo in the color code in table 1-- on CLAMBER dataset llama3.1 8B, performance of proposed method should not be highlighted for AUROC.   

(3) Some simple baselines are missing. On the detection task on open-source model, how does the performance compared to simple perplexity based detection? On the mitigation task, how does the performance compared to majority vote among the generated answers? -- this is particularly interesting since the design of detection score is to find consensus.


Decision: Reject