Text Ambiguity Score (TAS)

• Text Ambiguity Score (TAS) is a family of metrics that quantifies semantic ambiguity by measuring the diversity and spread among plausible interpretations.
• It employs multiple formulations, including interpretation-based entropy, embedding clustering, and path-kernel averaging, to capture uncertainty in text inputs.
• TAS is applied in various domains, such as Text-to-SQL systems and text-to-video retrieval, to trigger disambiguation, triage, and improve dataset quality.

The Text Ambiguity Score (TAS) is a family of information-theoretic and geometric metrics designed to quantify the semantic ambiguity present in a natural-language input or a discrete annotation distribution. Deployed in diverse applications—from clinical Text-to-SQL systems to text-to-video retrieval and soft-label annotation analysis—TAS measures the intrinsic uncertainty of an input by formalizing the diversity, spread, or conceptual distance among its plausible interpretations or labelings. By distinguishing input-driven ambiguity from downstream uncertainty, TAS enables targeted interventions such as clarification dialogues, triage, or dataset stratification in machine learning workflows.

1. Formal Definitions and Mathematical Foundations

TAS appears in several mathematically distinct but conceptually analogous forms, tailored to task structure.

Interpretation-based Entropy (CLUES framework):

Given an input query $q$, a set of $N$ generated interpretations $\mathcal{I} = \{I_1, ..., I_N\}$, and a pairwise semantic similarity kernel $k(I_i, I_j) \in [0,1]$, construct a similarity matrix $\mathbf{W}_{II}$, its degree matrix $\mathbf{D}_I$, and the graph Laplacian $\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$. A heat kernel is formed as $\mathbf{K}_I = \exp(-\tau \mathbf{L}_I)$ for temperature parameter $\tau>0$, normalized to a density matrix $\rho_I = \mathbf{K}_I/\mathrm{Tr}(\mathbf{K}_I)$. The TAS is the von Neumann entropy:

$N$0

This entropy surges when interpretations cluster into well-separated, semantically distinct groups (i.e., high ambiguity), and approaches zero when all readings are essentially equivalent (Ziletti et al., 12 Feb 2026).

Semantic Entropy over Embedding Clusters (UMIVR):

For text-to-video retrieval, let $N$1 be a query, $N$2 a corpus of captions, and $N$3, $N$4 their normalized embeddings. Retrieve the $N$5 captions nearest to $N$6, cluster into $N$7 groups, and define cluster probabilities:

$N$8

Semantic entropy is $N$9, yielding the normalized score:

$\mathcal{I} = \{I_1, ..., I_N\}$0

This value lies in $\mathcal{I} = \{I_1, ..., I_N\}$1, stratifying queries along an ambiguity axis (Zhang et al., 21 Jul 2025).

Concept-Wise Path Kernel Averaging (SAE framework):

Here, ambiguity is encoded as the average distance in the representation space of a sparse autoencoder (SAE):

$\mathcal{I} = \{I_1, ..., I_N\}$2

where $\mathcal{I} = \{I_1, ..., I_N\}$3 are two LLM-generated interpretations and $\mathcal{I} = \{I_1, ..., I_N\}$4 is a normalized path-kernel-induced concept distance (Hu et al., 16 May 2025).

Soft-Label Ambiguity (Quadratic Entropy with Abstentions):

Given a categorical annotation distribution $\mathcal{I} = \{I_1, ..., I_N\}$5, where $\mathcal{I} = \{I_1, ..., I_N\}$6 is the probability of the "can't solve" option, the score is:

$\mathcal{I} = \{I_1, ..., I_N\}$7

with $\mathcal{I} = \{I_1, ..., I_N\}$8. This construction asymmetrically penalizes irreducible ambiguity distinct from annotator confusion (Klugmann et al., 5 Oct 2025).

2. Algorithmic and Computational Procedures

TAS computation generally involves (i) generating candidate semantic variants, (ii) quantifying their divergence, and (iii) reducing the result to a scalar value.

Interpretation Entropy Algorithms:

Given small $\mathcal{I} = \{I_1, ..., I_N\}$9 (typically 2–4, e.g., for Text-to-SQL), core steps are:

• Generate interpretations via LLM or annotation;
• Evaluate semantic similarity $k(I_i, I_j) \in [0,1]$0 for all $k(I_i, I_j) \in [0,1]$1 (possible LLM-augmented equivalence prompts);
• Compute Laplacian and exponentiate to heat kernel;
• Normalize and compute von Neumann entropy from eigenvalues of $k(I_i, I_j) \in [0,1]$2.

Complexity is negligible for $k(I_i, I_j) \in [0,1]$3; low-rank methods address scalability.

Embedding Entropy Algorithms:

For text–video retrieval:

• Encode all captions and query;
• Retrieve top-$k(I_i, I_j) \in [0,1]$4 captions by cosine similarity;
• Cluster into $k(I_i, I_j) \in [0,1]$5 groups (e.g., K-means);
• Aggregate similarity mass and compute entropy over cluster probabilities;
• Normalize the entropy.

Thresholds (e.g., $k(I_i, I_j) \in [0,1]$6) select the regime triggering clarification.

Path-Kernel Averaging:

Given question $k(I_i, I_j) \in [0,1]$7 and interpretations $k(I_i, I_j) \in [0,1]$8, $k(I_i, I_j) \in [0,1]$9:

• Extract SAE activations per input;
• Approximate path kernel via interpolated gradients in autoencoder parameter space;
• Compute three pairwise distances (with suitable normalization);
• Average for final TAS.

Soft-label Ambiguity:

For $\mathbf{W}_{II}$0 categorical labels (possibly including abstentions):

• Compute empirical class probabilities $\mathbf{W}_{II}$1;
• Plug into $\mathbf{W}_{II}$2;
• Frequentist or Bayesian estimators handle bias and uncertainty quantification.

3. Interpretation, Theoretical Properties, and Protocols

TAS is always normed to a fixed range—either $\mathbf{W}_{II}$3, $\mathbf{W}_{II}$4, or $\mathbf{W}_{II}$5—enabling cross-task comparison. Key behaviors:

• Minimum TAS ($\mathbf{W}_{II}$6): All interpretations or neighbor captions collapse to a single semantic cluster or class, indicating unambiguous, sharply specified input.
• Maximum TAS: Interpretations or retrieved elements distribute uniformly across distinct clusters, marking maximal ambiguity.

Specific tasks operationalize TAS cutoffs:

• In CLUES, $\mathbf{W}_{II}$7 above median triggers clarification, while low $\mathbf{W}_{II}$8 proceeds directly to answer generation (Ziletti et al., 12 Feb 2026).
• In UMIVR, $\mathbf{W}_{II}$9 activates open-ended clarification; further intervention depends on subsequent reductions (Zhang et al., 21 Jul 2025).
• For annotation datasets, $\mathbf{D}_I$0 (moderate) or $\mathbf{D}_I$1 (high) guide review or curation (Klugmann et al., 5 Oct 2025).

Theoretical results demonstrate:

• TAS distinguishes ambiguity caused by genuine input uncertainty from that due to model instability or output variability when paired with instability scores (e.g., $\mathbf{D}_I$2 in CLUES).
• Path-kernel TAS, compared to embedding-only approaches, offers higher detection accuracy for ambiguous questions (e.g., 86.25% vs. 70–77.75%) (Hu et al., 16 May 2025).

4. Empirical Validations and Benchmarks

Empirical confirmation spans multiple domains:

Setting	Empirical Outcome	Reference
AmbigQA/SituatedQA	TAS $\mathbf{D}_I$3 enables regime separation, improving outcome prediction above baseline entropy of answers.	(Ziletti et al., 12 Feb 2026)
Clinical Text-to-SQL	High $\mathbf{D}_I$4, high $\mathbf{D}_I$5 regime contains 51% of errors but only 25% of queries, enabling focused triage.	(Ziletti et al., 12 Feb 2026)
Text-to-Video Retrieval	High initial TAS (e.g., 0.78) correlates with low Recall@1; clarification reduces TAS and boosts retrieval.	(Zhang et al., 21 Jul 2025)
AMBROSIA Benchmark	Path-kernel TAS: 86.25% detection accuracy; clear separation of ambiguous vs. unambiguous instance distributions	(Hu et al., 16 May 2025)
Annotation Stratification	Plug-in $\mathbf{D}_I$6 discriminates soft label ambiguity; Bayesian intervals inform credible region for ambiguity.	(Klugmann et al., 5 Oct 2025)

The consistent observation is that stratifying queries or instances by TAS enables more efficient downstream actions (clarification, review, or automatic acceptance), and that entropy-based and geometry-based TAS outperform standard embedding similarity metrics.

5. Relationship to Other Uncertainty and Instability Measures

TAS is conceptually orthogonal to model instability and mapping uncertainty measures.

• Instability Score ($\mathbf{D}_I$7, CLUES): Measures conditional diversity of outputs (e.g., SQL queries) after fixing an input interpretation; computed via heat-kernel entropy on the Schur complement of the semantic bipartite graph (Ziletti et al., 12 Feb 2026).
• Mapping Uncertainty Score (MUS, UMIVR): Quantifies text–video mapping ambiguity using Jensen-Shannon divergence; engaged after reducing semantic ambiguity via TAS (Zhang et al., 21 Jul 2025).
• Separation of Regimes: Only by decomposing total output uncertainty into TAS (ambiguity) and instability can systems distinguish cases requiring user disambiguation from those needing model improvement or fallback logic.

A monolithic uncertainty score (e.g., entropy on generated answers alone) cannot diagnose the root cause of output variability and thus conflates structurally different intervention regimes.

6. Statistical Inference, Thresholds, and Reporting

In annotation analysis, TAS supports both point estimation and full Bayesian inference.

• Frequentist estimator: $\mathbf{D}_I$8, $\mathbf{D}_I$9 = total, $\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$0 = abstentions; estimator is biased low but consistent; bias-corrected formulas available (Klugmann et al., 5 Oct 2025).
• Bayesian estimation: Placing a Dirichlet prior on class proportions yields posterior samples for $\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$1, allowing credible intervals for ambiguity estimation.
• Threshold selection: Empirically calibrated cutoffs ($\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$2, $\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$3) are recommended for moderate/high ambiguity; always accompany point estimates with measures of variance or credible intervals.

TAS thus provides not only a numeric score, but also the statistical infrastructure for principled downstream triage, active learning, and data quality analysis.

7. Practical Applications and Diagnostic Protocols

TAS is integrated in several modern NLP/NLU workflows:

• Interactive Query Clarification: UMIVR routes queries above a TAS threshold to open-ended clarification, leading to rapid entropy reduction and increased retrieval efficacy (Zhang et al., 21 Jul 2025).
• Failure Prediction and Triage: In clinical Text-to-SQL, queries with high TAS and/or high instability are triaged for clarification or human review, reducing the cost of pipeline errors (Ziletti et al., 12 Feb 2026).
• Ambiguity Detection and Disambiguation: In agentic tool-calling and API retrieval, path-kernel TAS is used to reliably flag ambiguous queries and trigger missing-concept prediction for more robust retrieval (Hu et al., 16 May 2025).
• Dataset Curation and Benchmarking: $\mathbf{L}_I = \mathbf{D}_I - \mathbf{W}_{II}$4 is leveraged to filter, stratify, and calibrate categorical datasets, supporting quality control and domain adaptation diagnostics (Klugmann et al., 5 Oct 2025).

This suggests that the explicit quantification of text ambiguity—achieved via diverse instantiations of TAS—increasingly defines best practices for robust, interactive, and transparent language-based systems. TAS provides the quantitative backbone for efficient clarification, principled statistical assessment, and the separation of true semantic ambiguity from downstream or model-induced uncertainty.