Definition

Given a set of points confined to a reduced coordinate system $y \in \mathbb{R}^r$ (e.g. the top-$r$ PCA coordinates of a local embedding neighbourhood), an implicit degree-$k$ polynomial surface is fit by minimizing the algebraic residual $p(y) = 0$ via total-least- squares (SVD on the design matrix of monomials up to degree $k$). At $k=2$ (quadric) the fit is classified into standard quadric types (ellipsoid, one/two-sheeted hyperboloid, paraboloid, cylinder, cone); the source paper reports the resulting type label but does not state the specific classification procedure used. A hyperboloid-type patch is one whose fitted quadric is classified as a one-sheeted hyperboloid: an open, saddle-curved surface, unlike a closed ellipsoid, and unlike the flat $k=1$ (affine) baseline.

Relative to manifolds-paraboloid and manifolds-affine

Paraboloid (circular × pinched continuum) is a different specific quadric shape (circular radius pinching to zero along an axial coordinate, established via PCA loop-radius inspection — a parametrization-mode claim) found in color hue/lightness representations. This node’s shape (one-sheeted hyperboloid, established via implicit polynomial surface fitting and total-least-squares residual minimization, with model selection between affine/quadric/cubic degrees) is a distinct quadric type found in sentence-embedding paraphrase neighbourhoods, and a distinct mode of specification (implicit level set vs. explicit parametrization) — both are quadric-family siblings under Manifolds, neither subsumes the other. Affine Subspace is the degree-1 baseline this node’s fits are compared against: the central finding is that the affine model is systematically outperformed by the curved quadric/cubic models, not merely comparable.

Key evidence

Bedratyuk (2026) generates large controlled families of near-paraphrase sentences (three template domains — institutional/policy, educational, medical — each with 4 lexical slots × 18 context-compatible variants, up to $18^4$ sentences per regime) and embeds them with sentence-transformers/all-mpnet-base-v2 (768-dim; robustness-checked against all-MiniLM-L6-v2, 384-dim). Reducing to an adaptively-sized PCA subspace (r = 10-37 dims, 90% explained-variance criterion), fitting implicit affine/quadric/cubic surfaces via total-least-squares shows a systematic, large residual drop with increasing degree — e.g. regime A-C5: affine RMSE 0.2296 → quadric 0.0165 → cubic 0.0082; B-C5: 0.1577 → 0.0142 → 0.0078; C-C5: 0.1174 → 0.0161 → 0.0098 — reproduced with the smaller MiniLM model (A-C5: 0.732 → 0.114; B-C5: 0.482 → 0.083; C-C5: 0.550 → 0.074). The fitted quadric carriers were classified, in almost all regimes, as one-sheeted-hyperboloid-type, not ellipsoidal — i.e. local paraphrase variation is better modeled as an open curved patch than a closed cluster. A downstream 18-class few-shot classification ablation (adding synthetic points generated by Newton-projection onto the fitted surface vs. linear interpolation vs. no-synthetic baseline) found no consistent accuracy/F1 improvement from the surface-based synthetic points (e.g. 10-shot: surface-based $\Delta$accuracy $=-0.0040$, $\Delta$macro-F1 $=+0.0010$) — a measured null causal-intervention result, reported alongside the shape claim.

How to detect it

Restrict to a tightly controlled local family of inputs (e.g. templated paraphrases varying one factor at a time); reduce to a PCA subspace retaining most variance; fit implicit polynomial surfaces of increasing degree (affine, quadric, cubic) via total-least-squares and compare held-out residual (RMSE/MAE); if quadric/cubic residuals are systematically and substantially lower than affine, classify the quadric’s type (ellipsoid vs. hyperboloid vs. paraboloid vs. cylinder) to determine whether the local patch is closed (ellipsoid) or open/saddle-shaped (hyperboloid).

Key papers

• Bedratyuk, L. (2026). Controlled Paraphrase Geometry in Sentence Embedding Space: Local Manifold Modeling and Latent Probing. arXiv:2605.01073 — origin of the local hyperboloid-quadric characterization of controlled paraphrase neighbourhoods.