Semantic Grounding Index: Geometric Bounds on Context Engagement in RAG Systems
This addresses the issue of unreliable outputs in RAG systems for users in AI and NLP, offering a theoretically grounded tool for detecting responses that need verification, though it is incremental as it builds on existing geometric and embedding methods.
The paper tackles the problem of hallucination in retrieval-augmented generation (RAG) systems by introducing the Semantic Grounding Index (SGI), a geometric measure based on angular distances in embedding space, and finds that hallucinated responses tend to stay close to questions rather than retrieved contexts, with effect sizes up to Cohen's d=2.05 and AUC improvements from 0.72 to 0.83.
When retrieval-augmented generation (RAG) systems hallucinate, what geometric trace does this leave in embedding space? We introduce the Semantic Grounding Index (SGI), defined as the ratio of angular distances from the response to the question versus the context on the unit hypersphere $\mathbb{S}^{d-1}$.Our central finding is \emph{semantic laziness}: hallucinated responses remain angularly proximate to questions rather than departing toward retrieved contexts. On HaluEval ($n$=5,000), we observe large effect sizes (Cohen's $d$ ranging from 0.92 to 1.28) across five embedding models with mean cross-model correlation $r$=0.85. Crucially, we derive from the spherical triangle inequality that SGI's discriminative power should increase with question-context angular separation $θ(q,c)$-a theoretical prediction confirmed empirically: effect size rises monotonically from $d$=0.61 -low $θ(q,c)$, to $d$=1.27 -high $θ(q,c)$, with AUC improving from 0.72 to 0.83. Subgroup analysis reveals that SGI excels on long responses ($d$=2.05) and short questions ($d$=1.22), while remaining robust across context lengths. Calibration analysis yields ECE=0.10, indicating SGI scores can serve as probability estimates, not merely rankings. A critical negative result on TruthfulQA (AUC=0.478) establishes that angular geometry measures topical engagement rather than factual accuracy. SGI provides computationally efficient, theoretically grounded infrastructure for identifying responses that warrant verification in production RAG deployments.