Geometric Suspicion in LLM Hallucination Detection

• Geometric Suspicion is a framework that assesses LLM responses by using convex hull geometry and archetypal analysis to measure semantic uncertainty.
• The method ranks individual responses through local metrics—density, consensus distance, and usage rarity—to identify potential hallucinations.
• Empirical results demonstrate that selecting the least suspicious responses via this framework reduces hallucination rates and improves F1 and AUROC scores.

Geometric Suspicion refers to the geometric analysis of semantic uncertainty in LLM outputs for the detection and reduction of hallucinations. It is realized as a black-box, rank-based criterion that attributes suspicion (or unreliability) at the level of individual responses, in contrast to conventional global uncertainty metrics. The framework underpinning Geometric Suspicion leverages archetypal analysis on response embeddings, convex hull volume calculations, and several local geometric statistics to achieve nuanced uncertainty quantification and practical hallucination reduction, with a theoretical justification linking convex volume and entropy (Phillips et al., 17 Sep 2025).

1. Geometric Framework for LLM Hallucination Uncertainty

The approach conceptualizes the model's response set to a given prompt as a finite point cloud in a reduced embedding space. Specifically, n sampled responses (r₁, ..., rₙ), including the default output, are mapped via a fixed sentence encoder and normalized PCA to X ∈ ℝ^{n × d'}. Archetypal Analysis (AA) is then employed on X, solving the bi-convex minimization problem

$\min_{\{B, A\}} \|X - XBA\|_F^2$

subject to simplex constraints: b_j ∈ Δₙ and a_i ∈ Δ_K. Here, B ∈ ℝ^{n×K} selects archetype locations Z = X B, and A ∈ ℝ^{K×n} encodes the convex representations of each response in terms of the archetypes. The archetypes {z₁, ..., z_K} effectively span the semantic extremes of the response set, defining a convex polytope hull. This forms the geometric substrate for both global (batch-level) and local (response-level) uncertainty metrics.

2. The Geometric Suspicion Metric

Geometric Suspicion addresses the attribution of local uncertainty by ranking individual model responses according to three geometric criteria, all operated in the batch embedding space:

• Local Density (L):

$$L(r_i) = \frac{1}{k} \sum_{x_j \in N_k(x_i)} \| x_i - x_j \|_2$$

where N_k(x_i) denotes the k-nearest neighbors of x_i. High values imply semantic isolation, increasing suspicion.

• Distance from Consensus (D):

$$D(r_i) = \| x_i - x_c \|_2, \quad x_c = \frac{1}{n} \sum_{j} x_j$$

Measures deviation from batch consensus; distant responses are deemed less reliable.

• Usage Rarity (U):

$U(r_i) = \sum_{k=1}^{K} A_{ik}(1 - \bar{A}_k)$

where A_{ik} is the archetype coefficient for r_i, and $\bar{A}_k = \frac{1}{n}\sum_j A_{jk}$ is the mean weight for archetype k. This captures the extent to which r_i is constructed from rarely used archetypes, implicating atypicality.

Each response is ranked on each criterion; the composite Geometric Suspicion score is the sum of these ranks: $$S(r_i) = \operatorname{rank}_L(r_i) + \operatorname{rank}_D(r_i) + \operatorname{rank}_U(r_i)$$ with lower S(r_i) indicating higher reliability.

3. Global Versus Local Uncertainty

The framework distinctly separates global and local uncertainty estimation:

• Global uncertainty is captured by the Geometric Volume:

$$H_G(X) = \log(\operatorname{vol}(\operatorname{conv}(Z)) + \varepsilon)$$

where $\operatorname{conv}(Z)$ is the convex hull of archetypes and ε is a regularization constant for stability. Large H_G(X) suggests higher semantic diversity in responses and, empirically, a greater likelihood of hallucination at the batch level.

• Local uncertainty is assigned by Geometric Suspicion S(r_i), providing a response-wise suspicion ranking. This enables selective response curation—e.g., a Best-of-N strategy selecting the least suspicious candidate—as opposed to rejecting an entire batch when global uncertainty is high.

The capability to attribute uncertainty both globally (batch-wise, via convex dispersion) and locally (response-wise, via geometric suspicion) distinguishes this method from prior black-box or white-box approaches, most of which yield only a singular global (dispersion or entropy) estimate or require internal model access.

4. Archetypal Analysis and Geometric Volume

Archetypal Analysis is central for identifying the convex hull's semantic “corners.” Each archetype zₖ is a convex combination of observed responses, lying on or near the hull boundary. The convex hull volume of these archetypes serves as a sensitive indicator of global variability and supports batch-level uncertainty calculation.

Moreover, the archetypal decomposition enables the computation of usage rarity at the local level: the archetype allocation coefficients determine if a response is attributed to seldom-used extremes (which may signal hallucination or semantic drift).

The direct computation of the convex hull volume in PCA space for K archetypes yields a global uncertainty score grounded in the actual boundaries of observed response diversity.

5. Empirical Results and Hallucination Mitigation

Experiments evaluated the framework on benchmark datasets including CLAMBER, TriviaQA, ScienceQA, MedicalQA, and K-QA. Geometric Volume achieved F1 and AUROC metrics comparable to, or exceeding, state-of-the-art semantic entropy and volume-based baselines, especially on medical QA datasets where hallucinations may have severe consequences.

For hallucination reduction, applying Geometric Suspicion for Best-of-N response selection robustly reduced hallucination rates compared to default response outputs. This effect was consistent across several LLMs of varying architectures and sizes (e.g., GPT-3.5-Turbo, GPT-4o Mini, Llama3.1-8b, Qwen3-8b).

6. Theoretical Foundations: Volume–Entropy Link

The geometric approach is theoretically justified via a result linking convex hull volume to differential entropy: if the response embeddings are distributed within a simplex Δ spanned by the archetypes, then

$$H(x) = -\int_\Delta p(x) \log p(x)\, dx \leq \log V$$

where V is the volume of the simplex. Equality holds when p(x) is uniform on Δ. Thus, log-volume provides an upper bound on semantic entropy, formally connecting the Geometric Volume metric to information-theoretic uncertainty and substantiating its use as a surrogate for entropy in black-box settings.

Summary Table: Geometric Suspicion Framework

Aspect	Global Uncertainty	Local Uncertainty (Geometric Suspicion)
Estimator	log-volume of archetypes	Composite rank (density, consensus, rarity)
Implementation	Archetypal analysis, PCA	kNN, centroid distance, usage rarity
Output	Batch-level score	Response-level suspicion, enables Best-of-N selection

This comprehensive geometric framework advances hallucination detection and reduction in LLMs by integrating the semantics-aware convex hull geometry of model outputs with both global and local suspicion quantification, supported by strong theoretical and empirical evidence (Phillips et al., 17 Sep 2025).