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arxiv: 2512.04351 · v3 · submitted 2025-12-04 · 💻 cs.LG

Distance Is All You Need: Radial Dispersion for Uncertainty Estimation in Large Language Models

Manh Nguyen , Sunil Gupta , Hung Le This is my paper

Pith reviewed 2026-05-17 01:37 UTC · model grok-4.3

classification 💻 cs.LG
keywords uncertainty estimationhallucination detectionlarge language modelsradial dispersionembedding spacemodel-agnostictraining-freefree-form QA
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The pith

A simple count of how far sampled LLM outputs sit from their average embedding detects hallucinations more reliably than complex clustering methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to show that uncertainty in large language models can be read directly from the geometric spread of their generated answers when those answers are turned into vectors on the unit sphere. Instead of relying on semantic clustering or peeking inside the model, the authors measure the total L1 distance of each embedding from the average of the set. If this radial dispersion tracks semantic variability, then a lightweight score built from it should flag hallucinations without extra machinery. The claim matters because current uncertainty tools are often brittle or expensive, so a purely geometric alternative could make reliable detection practical at scale. When the method is applied to free-form question answering, it reaches state-of-the-art detection rates while staying stable across different numbers of samples and different embedding models.

Core claim

The central claim is that the Radial Dispersion Score, defined as the sum of L1 distances between N sampled generation embeddings and their empirical centroid on the unit hypersphere, supplies a direct geometric signal of semantic variability; a probability-weighted version of the same score further improves performance when token probabilities are available, and both versions deliver state-of-the-art hallucination detection across four challenging free-form QA datasets and four different LLMs while remaining robust to sample size and embedding choice.

What carries the argument

Radial Dispersion Score (RDS), which sums the L1 distances of unit-hypersphere embeddings from their empirical mean to quantify dispersion without clustering or internal access.

If this is right

  • RDS achieves state-of-the-art hallucination detection on four free-form QA datasets using four different LLMs.
  • Performance stays stable when the number of samples or the embedding model is changed.
  • The probability-weighted variant improves results whenever token probabilities can be obtained.
  • The same dispersion measure supplies lightweight per-sample uncertainty estimates that work alongside existing probability or consistency checks.
  • Because the method needs no training and no model internals, it applies immediately to new models and tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If dispersion alone suffices, then many uncertainty problems may reduce to measuring spread in a fixed embedding space rather than building separate semantic modules.
  • The approach could be tested on structured tasks such as code generation or mathematical reasoning to see whether the same geometric signal remains informative.
  • Combining radial dispersion with existing consistency checks might produce hybrid estimators whose error rates fall below either method alone.
  • The method's independence from model internals suggests it could serve as an external audit tool for proprietary LLMs where only outputs are observable.

Load-bearing premise

That the geometric spread of embeddings from their average directly and reliably reflects semantic variability and therefore model uncertainty.

What would settle it

A dataset in which generations are semantically consistent and factually correct yet produce large radial dispersion scores, or vice versa, would show the metric does not track uncertainty as claimed.

Figures

Figures reproduced from arXiv: 2512.04351 by Hung Le, Manh Nguyen, Sunil Gupta.

Figure 1
Figure 1. Figure 1: RDS vs. EigenEmbed across three uncertainty regimes illustrated using ten unit-norm 2D embeddings. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ablation on the number of sampled responses [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ablation on the number of sampled responses [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Detecting uncertainty in large language models (LLMs) is essential for building reliable systems, yet many existing approaches are overly complex and depend on brittle semantic clustering or access to model internals. We introduce Radial Dispersion Score (RDS), a simple, training-free, fully model-agnostic uncertainty metric that measures the radial dispersion of sampled generations in embedding space. Specifically, given $N$ sampled generations embedded on the unit hypersphere, RDS computes the total l1 distance from the empirical centroid, i.e., the mean embedding, providing a direct geometric signal of semantic variability. A lightweight probability-weighted variant further incorporates the model's own token probabilities when available, outperforming nine recent state-of-the-art baselines. Moreover, RDS naturally extends to effective per-sample uncertainty estimates that complement probability- and consistency-based methods while remaining lightweight for practical use. Across four challenging free-form question-answering datasets and four LLMs, our metrics achieve state-of-the-art hallucination detection performance, while remaining robust and scalable with respect to sample size and embedding choice. These results highlight the practical value of RDS and its contribution toward improving the trustworthiness of LLMs. Code is publicly available at https://github.com/manhitv/RDS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Radial Dispersion Score (RDS), a training-free, model-agnostic uncertainty metric for LLMs that computes the total L1 distance of N sampled generation embeddings from their empirical centroid on the unit hypersphere. A probability-weighted variant is also proposed. The central claim is that RDS (and its variant) achieves state-of-the-art hallucination detection performance across four free-form QA datasets and four LLMs while remaining robust to sample size and embedding choice.

Significance. If the empirical claims hold after verification, RDS offers a lightweight geometric alternative to clustering-based or internal-state methods for uncertainty estimation. Public code release aids reproducibility. The approach could support practical hallucination detection in deployed LLM systems if it reliably separates semantic variability from other sources of dispersion.

major comments (2)
  1. [RDS Definition] RDS definition (as stated in the abstract): the claim that total L1 distance from the empirical centroid provides a 'direct geometric signal of semantic variability' does not address multi-modal generation distributions. In free-form QA, distinct valid or invalid answers produce embeddings in separate regions; their centroid lies near the origin, inflating RDS regardless of whether the spread reflects true uncertainty or answer diversity. No controls or analysis are provided to show the metric distinguishes these cases without clustering or model internals.
  2. [Experimental Results] Experimental results (as summarized in the abstract): the manuscript reports SOTA hallucination detection performance yet provides no details on statistical tests, exact baseline implementations, data splits, or potential post-hoc choices. This absence makes it impossible to assess whether the reported gains are robust or reproducible, directly undermining the central empirical claim.
minor comments (1)
  1. [Method] Clarify the precise embedding normalization procedure and any distance function variants beyond L1 in the method description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to improve clarity and completeness.

read point-by-point responses

  1. Referee: [RDS Definition] RDS definition (as stated in the abstract): the claim that total L1 distance from the empirical centroid provides a 'direct geometric signal of semantic variability' does not address multi-modal generation distributions. In free-form QA, distinct valid or invalid answers produce embeddings in separate regions; their centroid lies near the origin, inflating RDS regardless of whether the spread reflects true uncertainty or answer diversity. No controls or analysis are provided to show the metric distinguishes these cases without clustering or model internals.

    Authors: We appreciate the referee's observation regarding the potential limitations of the centroid-based approach in multi-modal settings. Indeed, when generations form distinct clusters corresponding to different valid or invalid answers, the empirical centroid can approach the origin, resulting in elevated RDS values that may reflect answer diversity rather than uncertainty per se. However, our empirical results across multiple datasets and models demonstrate that RDS still effectively correlates with hallucination labels, suggesting that in practice, high dispersion often indicates cases where the model is uncertain about the correct response. To directly address this concern, we will revise the manuscript to include a new subsection discussing multi-modal distributions. This will incorporate visualizations of embedding distributions for sample cases with multiple modes and a comparison of RDS performance on subsets identified as multi-modal versus unimodal. We will also clarify the scope of our claims and note that RDS is intended as a lightweight complement to more sophisticated clustering methods. revision: yes

  2. Referee: [Experimental Results] Experimental results (as summarized in the abstract): the manuscript reports SOTA hallucination detection performance yet provides no details on statistical tests, exact baseline implementations, data splits, or potential post-hoc choices. This absence makes it impossible to assess whether the reported gains are robust or reproducible, directly undermining the central empirical claim.

    Authors: We acknowledge that the current manuscript lacks sufficient details on the experimental setup to ensure full reproducibility and robustness assessment. In the revised version, we will expand the experimental section to include: (1) details on statistical significance testing, such as the use of paired t-tests or bootstrap methods to compare AUC scores; (2) precise descriptions of baseline implementations, including any specific hyperparameters, libraries, and code references; (3) information on data splits, noting that our evaluations are zero-shot on the full test sets of the four QA datasets without any training or validation splits; and (4) confirmation that no post-hoc selection of results was performed, with all experiments run under fixed protocols. Additionally, we will make the complete experimental code, including scripts for baseline reproduction, publicly available alongside the existing repository to facilitate verification. revision: yes

Circularity Check

0 steps flagged

RDS defined directly from embeddings; no circular derivation or self-referential reduction

full rationale

The paper introduces RDS as an explicit, training-free definition: given N sampled generations embedded on the unit hypersphere, RDS is the total L1 distance from the empirical centroid (mean embedding). This is a direct geometric computation using a fixed distance function and does not reduce to a fitted parameter, self-referential equation, or load-bearing self-citation. No uniqueness theorems, ansatzes smuggled via prior work, or renamings of known results are invoked in the central construction. Performance results are empirical evaluations across datasets and models rather than derivations that loop back to inputs by construction. The method remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard embedding assumptions and the new geometric definition; no free parameters, invented entities, or ad-hoc axioms beyond domain conventions are introduced.

axioms (2)
  • domain assumption Sampled generations can be embedded as points on the unit hypersphere
    Invoked to enable the radial dispersion calculation via L1 distance from the centroid.
  • domain assumption Dispersion in embedding space corresponds to semantic variability and uncertainty
    Core premise linking the geometric score to the practical goal of hallucination detection.

pith-pipeline@v0.9.0 · 5512 in / 1298 out tokens · 69262 ms · 2026-05-17T01:37:30.575802+00:00 · methodology

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Reference graph

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6 extracted references · 6 canonical work pages · 2 internal anchors

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