Geometry of Semantic Space: Comparative Study of Discrete and Continuous Models

Semantic type mismatch between a noun and its context is central to coercion phenomena. This paper introduces a graph-based method to examine how lexical and contextual type information is reflected in word embeddings. We select nouns from ten semantic types, annotate corpus instances for type matching (matching vs. coercion vs. other mismatch vs. unrestricted), and construct graphs using BERT and sense-enhanced embeddings. Two metrics -- Neighbor Type Probability (NTP) and Neighbor Type Entropy (NTE) -- are proposed to analyze neighborhood type distributions. Results show that graphs constructed with sense-enhanced embeddings reflect semantic type information better, and matching and mismatch sentences can be distinguished through the proposed metrics.

When it comes to generating vector representations of words, current language models are achieving high-quality results. However, what is not known is the extent to which knowledge about semantic relations is represented in the geometry of the semantic spaces created in this way. In order to answer this question, we study the relation geometry of such semantic spaces from three perspectives. We first examine whether words standing in a particular relation to a target word~(called relata) occupy the same region in semantic space, and whether the regions corresponding to different relations are distinct from each other. We then verify to what extent semantic spaces reflect certain well-known properties of relations, such as symmetry, asymmetry, and transitivity. Finally, we consider which information about the target words and relata is more important for relation geometry: their surface forms, or their contexts. We conduct experiments on six semantic relations using causal, masked, and diffusion language models. The results show that relata in asymmetric relations relatively clearly occupy a distinct region in semantic space. Asymmetric relations' properties are only moderately well encoded in the semantic space, yet better than those of symmetric ones. Furthermore, when considering the question which information source has the strongest impact on results amongst the models we evaluated, we find that lexical information tends to be more important for the causal language model, whereas contextual information is more important for the masked and diffusion language models. Our results empirically show that relation geometry is not equally well-represented for all relations in semantic space, suggesting that there is a difference in how well semantic relations might be learned from distributional information alone.

The paper studies the local geometry of embedding clouds induced by \emph{controlled local classes of semantically close sentences}. The central question is how controlled paraphrase-like semantic variation is organized in sentence embedding space and whether this local structure can be explicitly modeled by low-degree fitted carriers. We introduce a local geometric modeling scheme based on affine, quadratic, and cubic fitted models. We also use a surface-based latent probing procedure that constructs synthetic latent points in a reduced local PCA space with respect to the fitted carrier. The procedure is intended as an offline method for representation-space analysis, local manifold modeling, and geometry-aware latent probing. Generated latent points are evaluated using criteria that measure consistency with the fitted surface, preservation of neighborhood structure, agreement with the empirical distribution, stability of Hessian-based second-order shape descriptors, and stability of fitted-model coefficients. Experiments on controlled sets of semantically close sentences show that nonlinear local models describe embedding clouds more accurately than affine models. Surface-based generation provides strong fitted-geometry fidelity, including surface consistency, Hessian-based shape consistency, and coefficient consistency. Downstream experiments show that geometric validity of synthetic latent points does not automatically translate into improved classification performance. The results support explicit local geometric modeling of sentence embedding space and highlight the need to distinguish geometric validity from discriminative utility. As a resource contribution, we introduce \textbf{CoPaGE-300K}, a controlled template-based dataset of semantically close sentence variants with slot-level annotations and precomputed sentence embeddings.