Geometric Signatures of Semantic Entailment in Probabilistic Embedding Spaces

Semantic entailment and information containment are inherently asymmetric: a specificstatement entails a more general one, but not vice versa. However, modern text embeddingmodels represent meaning as points in a metric space, relying on symmetric similarity measuresthat fundamentally fail to capture this asymmetry and the uncertainty inherent in naturallanguage.We propose a probabilistic embedding framework that models text representations as Gaus-sian distributions rather than deterministic vectors. This formulation enables two critical capa-bilities absent from standard embeddings: (i) explicit modeling of semantic uncertainty throughlearned variance, and (ii) directional similarity via asymmetric divergence measures. Using Kull-back–Leibler divergence between embeddings, we show that semantic entailment emerges as aform of distributional containment, where broader queries geometrically enclose more specificdocuments.Empirically, we demonstrate that entailment pairs exhibit a consistent geometric signature:queries have systematically higher variance than their entailed documents, and forward KLdivergence from document to query is significantly lower than the reverse. Across multiplebenchmarks, over 86% of entailment pairs cluster in a distinct region of a joint variance–KLasymmetry space, while non-entailment pairs occupy complementary regions, indicating robuststructural separation in the learned embedding geometry.Beyond representation analysis, we show that learned uncertainty has direct practical util-ity. Query variance strongly correlates with retrieval difficulty, enabling an uncertainty-awareadaptive retrieval strategy that dynamically adjusts the number of retrieved candidates. Thisapproach achieves higher recall than fixed-budget retrieval while reducing average computa-tion, demonstrating that probabilistic embeddings support both semantic reasoning and efficientsystem-level decision making.Our results suggest that probabilistic embeddings provide a principled geometric foundationfor modeling semantic asymmetry, bridging representation learning, entailment reasoning, anduncertainty-aware retrieval.