Category-level Geometry Embedding (CGE)

• Category-level Geometry Embedding (CGE) is a representation that organizes dense 3D geometric descriptors alongside semantic channels to capture intra-category regularities.
• It fuses multi-view RGB-D observations, DINOv2 feature maps, and category prototypes to enable robust applications in robotic manipulation, pose estimation, and segmentation.
• CGE supports task-conditioned semantic shape reconstruction and embedding-space analysis, addressing challenges like occlusion, domain invariance, and expert structure recovery.

Searching arXiv for the cited papers to ground the article in current records. I'll look up the cited arXiv records and use them as the research basis for the article. Searching for "(Wang et al., 2024) GenDP 3D Semantic Fields for Category-Level Generalizable Diffusion Policy". Category-level Geometry Embedding (CGE) denotes a representational pattern in which geometry is organized at the level of semantic categories rather than only individual instances or entire domains. Across robotic manipulation, category-level pose estimation, domain-generalized 3D semantic segmentation, evaluation of text embedding spaces, and geometric relational representation learning, CGE is used to couple geometric structure with category semantics, prototypes, or expert-defined relations so that the learned space preserves intra-category regularities and inter-category structure (Wang et al., 2024, Li et al., 6 Feb 2025, He et al., 15 Aug 2025, Zhu et al., 22 Jun 2026, Xiong, 2024).

1. Conceptual scope and formal definitions

In the manipulation setting of "GenDP: 3D Semantic Fields for Category-Level Generalizable Diffusion Policy" (Wang et al., 2024), CGE is instantiated as a dense 3D representation that ties point-level geometry to category-consistent semantics. A 3D descriptor field is defined as

$F_{\text{desc}} : \mathbb{R}^3 \to \mathbb{R}^F,$

and a 3D semantic field is defined as

$S : \mathbb{R}^3 \to \mathbb{R}^M.$

For a point cloud $\mathcal{P} = \{p_i\}_{i=1}^K$, the descriptor field is represented as

$$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$

and the semantic field as

$$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$

with cosine-similarity channels

$$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$

The resulting embedding can be written as

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$

This formulation is explicitly dense in 3D, explicitly category-level, and explicitly joint in geometry and semantics (Wang et al., 2024).

In "GCE-Pose: Global Context Enhancement for Category-level Object Pose Estimation" (Li et al., 6 Feb 2025), CGE is framed as a canonical representation of category-level shape and semantics that remains usable under partial visibility. The central object is a reconstructed semantic shape

$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$

built from a category-specific geometric prototype, a category-level semantic prototype, deformation bases, and scale: $$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$ The paper explicitly states that the core Category-level Geometry Embedding is the dense global feature $\mathbf{F}_{\text{global}}$, which combines global geometry and category-level semantics in canonical space (Li et al., 6 Feb 2025).

In the 3D segmentation framework "Domain-aware Category-level Geometry Learning Segmentation for 3D Point Clouds" (He et al., 15 Aug 2025), CGE is defined as a geometry tensor

$S : \mathbb{R}^3 \to \mathbb{R}^M.$0

where $S : \mathbb{R}^3 \to \mathbb{R}^M.$1 are backbone point features and $S : \mathbb{R}^3 \to \mathbb{R}^M.$2 is a global geometry embedding matrix constructed from class-specific geometry projection matrices. Here CGE is not a pointwise semantic field but a category-indexed geometry basis shared across the scene and coupled to semantic learning through a dedicated loss (He et al., 15 Aug 2025).

In "Do LLM Embedding Spaces Recover Expert Structure?" (Zhu et al., 22 Jun 2026), CGE is not introduced as a named module but is conceptually the geometry induced by category prototypes in an LLM embedding space. For category $S : \mathbb{R}^3 \to \mathbb{R}^M.$3,

$S : \mathbb{R}^3 \to \mathbb{R}^M.$4

and the model representational dissimilarity matrix is

$S : \mathbb{R}^3 \to \mathbb{R}^M.$5

The question is whether the resulting category geometry recovers an externally specified expert structure rather than only domain or stylistic separation (Zhu et al., 22 Jun 2026).

In the dissertation "Geometric Relational Embeddings" (Xiong, 2024), the CGE idea appears as a general geometric-relational program in which categories, concepts, or labels are embedded as geometric objects such as boxes, balls, hyperplanes, or cones; entities are embedded as points; and category relations are expressed through containment, overlap, disjointness, or transformation constraints. This suggests a broad definition of CGE: a representation in which category structure is encoded geometrically rather than only by class logits or unstructured vectors (Xiong, 2024).

2. Category-level geometry in robotic manipulation

GenDP constructs CGE from multi-view RGB-D observations and DINOv2 descriptors. The per-time-step input is a set of $S : \mathbb{R}^3 \to \mathbb{R}^M.$6 RGB-D observations

$S : \mathbb{R}^3 \to \mathbb{R}^M.$7

For each RGB image $S : \mathbb{R}^3 \to \mathbb{R}^M.$8, DINOv2 produces a dense feature map

$S : \mathbb{R}^3 \to \mathbb{R}^M.$9

For a 3D point $\mathcal{P} = \{p_i\}_{i=1}^K$0, the method projects $\mathcal{P} = \{p_i\}_{i=1}^K$1 into each image, samples features $\mathcal{P} = \{p_i\}_{i=1}^K$2, reads depth $\mathcal{P} = \{p_i\}_{i=1}^K$3, computes the depth difference $\mathcal{P} = \{p_i\}_{i=1}^K$4, and fuses the multi-view features via a weighted fusion function

$\mathcal{P} = \{p_i\}_{i=1}^K$5

In practice, the implementation uses D$\mathcal{P} = \{p_i\}_{i=1}^K$6Fields to extract a point cloud $\mathcal{P} = \{p_i\}_{i=1}^K$7 and descriptors $\mathcal{P} = \{p_i\}_{i=1}^K$8, and it weights views where $\mathcal{P} = \{p_i\}_{i=1}^K$9 is small (Wang et al., 2024).

Semantic channels are then built from reference descriptors. Several 2D reference images are collected for the target category and the parts relevant to the task. DINOv2 is applied to these images, feature vectors for task-relevant parts are selected and averaged, and the resulting reference descriptors $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$0 define semantic parts such as shoe tongue, shoe cuff, knife handle, mug handle, or can tab. For each 3D point descriptor $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$1, cosine similarity to each reference gives

$$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$2

so that each semantic channel becomes a scalar field over 3D space indicating similarity to a task-relevant part (Wang et al., 2024).

The final policy input is the point feature

$$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$3

which is processed by PointNet++ into a global visual feature vector. Geometry is therefore encoded by the point cloud $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$4, by depth consistency weighting using $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$5, and by multi-view fusion $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$6. Semantics are encoded by DINOv2 features, reference descriptors, and cosine-similarity channels. The paper explicitly states that geometry provides where and semantics provide what/which part (Wang et al., 2024).

This representation is designed to resolve geometric ambiguity and to expose subtle details. The reported examples include marker head versus tail, knife blade versus handle, similar-looking sides of a pen or toothbrush, toothbrush head versus handle, soda can tab versus bulk of the can, and shoes’ tongue versus cuff. The paper states that visualizations show localized heatmaps highlighting book titles, can tabs, shoelaces, and mug handles, and that these patterns are category-consistent across instances (Wang et al., 2024). A plausible implication is that, in this formulation, CGE is not merely a shape descriptor but a task-conditioned semantic geometry field.

3. Canonical prototypes and global context in pose estimation

GCE-Pose studies the model-free category-level pose setting in which the specific CAD model of the test instance is unavailable. The input is a single RGB-D frame and a category mask, from which the method obtains a partial RGB image $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$7 and a partial point cloud $$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$8. The target is the 9DoF pose and size

$$\mathcal{F} = \{ \mathcal{F}_i \in \mathbb{R}^F \mid i = 1,\dots,K\},$$9

Its solution is explicitly based on category-level global context priors (Li et al., 6 Feb 2025).

For each category $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$0, the method learns a geometric prototype

$$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$1

and a category-level semantic prototype

$$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$2

Semantic prototype construction uses multiple rendered RGB-D views, DINOv2 feature extraction, 3D lifting with

$$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$3

aggregation by $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$4-nearest neighbors, and averaging over all training instances: $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$5 The result is a canonical semantic embedding aligned point-wise with the geometry of the category mean shape (Li et al., 6 Feb 2025).

The deep Linear Semantic Shape Model defines a low-dimensional family of semantic shapes: $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$6 A key modeling choice is stated directly: semantics are kept attached to their points during deformation, so all shapes in the family share the same semantic prototype $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$7 while geometry deforms (Li et al., 6 Feb 2025). This yields a canonical index space in which geometry varies but semantic correspondence remains fixed.

Training proceeds in two stages. Stage 1 fits the linear model to full shapes using a Chamfer-distance reconstruction loss

$$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$8

Stage 2 trains a deformation network $$\mathcal{C} = \{ \mathcal{C}_i \in \mathbb{R}^M \mid i = 1,\dots,K\},$$9 and a scale network $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$0 to infer deformation coefficients and scale from partial observations, with the prototype and bases frozen and with parameter regularization

$$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$1

combined as

$$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$2

The stated purpose is to complete a partial point cloud into a full semantic shape consistent with the learned category prior (Li et al., 6 Feb 2025).

The Global Context Enhanced fusion module then injects this category-level geometry embedding into the local RGB-D pose backbone. Partial point features $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$3, DINOv2 semantic features $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$4, cross-attention keypoints, and the Geometric-Aware Feature Aggregation module produce keypoint features $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$5. In parallel, SSR produces reconstructed points $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$6 and global features $$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$7. Positional embeddings, shared-space projections, and cross-attention yield

$$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$8

The enhanced representation is used to regress keypoint NOCS coordinates and then pose and size (Li et al., 6 Feb 2025).

The paper directly ties robustness under occlusion to this global context prior and reports performance on HouseCat6D and NOCS-REAL275. It states that GCE-Pose performs semantic shape reconstruction with a category prior, and that the global context enhanced fusion module effectively fuses partial RGB-D observations and reconstructed global context for novel instances under partial visibility (Li et al., 6 Feb 2025). This suggests a CGE regime in which missing geometry is not inferred purely from local evidence but through a category-conditioned completion model.

4. Category-level geometry as domain-invariant structure in 3D segmentation

In domain-generalized 3D semantic segmentation, the category-level geometry learning framework argues that existing methods learn global geometric patterns of the overall domain while ignoring category-level distribution and alignment. The formulation assumes a source domain and unseen target domains with

$$\mathcal{C}_{ij} = \frac{\mathcal{F}_i \cdot \mathcal{F}_{\text{ref},j}}{\|\mathcal{F}_i\|\, \|\mathcal{F}_{\text{ref},j}\|}.$$9

and a standard segmentation objective

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$0

The CGE contribution is to explicitly model category-specific geometry as an auxiliary structure rather than relying only on augmentation or global shape statistics (He et al., 15 Aug 2025).

The overall pipeline begins with a point cloud $$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$1 and backbone features

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$2

Category-level geometry construction defines a mapping

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$3

where $$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$4 concatenates class-specific geometry projection matrices

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$5

The class-specific optimal geometric features $$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$6 are constructed in Wasserstein space through an entropic optimal transport formulation solved with Sinkhorn divergence: $$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$7 The resulting tensor

$$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$8

contains, for each point, an $$E(p_i) = \bigl[p_i,\ S(p_i)\bigr] \in \mathbb{R}^{3+M}.$$9-dimensional geometric descriptor for each semantic category (He et al., 15 Aug 2025).

Coupling of geometry and semantics is implemented through a correlation matrix

$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$0

and the geometry property learning loss

$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$1

The paper states that this both learns the relationship between geometry attributes and categories and backpropagates through $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$2 and $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$3 into the backbone, forcing the backbone to produce features whose geometry embeddings are discriminative at the category level (He et al., 15 Aug 2025).

Because $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$4 is batch-dependent, stability is imposed through a momentum update

$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$5

where $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$6 is computed from reliable points. The paper states that $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$7 becomes a running geometric basis updated slowly over training and based mostly on correctly classified points (He et al., 15 Aug 2025).

Geometric Consistent Learning extends this by simulating latent 3D distributions through Physics-inspired Adverse Geometry Simulation and embedding the simulated features in the same geometry basis: $\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$8 The corresponding loss

$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k)$9

forces consistent category-level geometric embeddings across original and simulated domains (He et al., 15 Aug 2025). The full optimization is

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$0

The reported empirical pattern is that CGE alone improves mIoU from 31.4% to 33.0% on SemanticKITTI $$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$1 SemanticSTF, while adding GCL raises it to 37.5%, and test-time augmentation reaches 39.1% (He et al., 15 Aug 2025). The paper also reports that removing geometry awareness in Wasserstein space lowers mean mIoU from 37.5% to 35.2%, and removing alignment lowers it to 33.1% (He et al., 15 Aug 2025). Within this literature, CGE is therefore treated as a domain-invariant geometric prior indexed by semantic category.

5. Prototype geometry and expert structure in embedding-space analysis

In the study of LLM embeddings, CGE appears as a question about whether prototype geometry over categories recovers an externally defined expert structure. The dataset contains 28 Reddit communities, including 17 mental-health categories and 11 controls. Each category $$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$2 is assigned a binary symptom profile

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$3

with $$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$4 expert-informed symptom dimensions. The expert matrix is

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$5

and the expert representational dissimilarity matrix is

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$6

where $$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$7 is Jaccard distance. Control communities are assigned the all-zero vector by construction (Zhu et al., 22 Jun 2026).

Model geometry is built from L2-normalized document embeddings and category prototypes

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$8

The induced category geometry is the model RDM

$$\mathbf{U}_k = (\mathbf{X}_k, \mathbf{F}_k) = \bigl( s^k \odot \bigl(c^k + \sum_{i=1}^D a_i^k v_i^k\bigr),\; c^k_{\text{sem}} \bigr).$$9

with cosine distance. Representational Similarity Analysis then compares the expert and model geometries through

$\mathbf{F}_{\text{global}}$0

using the upper-triangular entries of the RDMs (Zhu et al., 22 Jun 2026).

The paper reports that on the MH-only 17-by-17 subset, pretrained embeddings already show substantial alignment with the expert reference, with $\mathbf{F}_{\text{global}}$1 for Qwen3-Embedding-0.6B and $\mathbf{F}_{\text{global}}$2 for Qwen3-Embedding-4B. After supervised fine-tuning, the values rise to $\mathbf{F}_{\text{global}}$3 and $\mathbf{F}_{\text{global}}$4, respectively (Zhu et al., 22 Jun 2026). On the full 28-by-28 set, the reported values are 0.703 and 0.735 zero-shot, increasing to 0.758 and 0.789 after fine-tuning; partial RSA controlling for a domain RDM yields 0.567 and 0.609 zero-shot, increasing to 0.651 and 0.682 after fine-tuning (Zhu et al., 22 Jun 2026). The stated interpretation is that alignment is level-dependent and that high category separability does not imply recovery of expert-defined structure.

The paper supplements global geometry with local prototype analyses. Typicality is defined as

$\mathbf{F}_{\text{global}}$5

and category-level cohesion as

$\mathbf{F}_{\text{global}}$6

A prototype margin is defined by

$\mathbf{F}_{\text{global}}$7

and pairwise boundary ambiguity is quantified via

$\mathbf{F}_{\text{global}}$8

with

$\mathbf{F}_{\text{global}}$9

The boundary alignment score

$S : \mathbb{R}^3 \to \mathbb{R}^M.$00

measures whether residual ambiguity is concentrated among expert-neighbor pairs rather than non-neighbors (Zhu et al., 22 Jun 2026).

This work also provides a confound-sensitive interpretation of CGE. The reported confound RDMs include VAD, LIWC affect+social, lexical style, and ETM topic distribution. The paper states that VAD is not a strong confound, while ETM topic structure is the strongest confound; however, even controlling for ETM topic structure, residual expert alignment remains substantial and significant (Zhu et al., 22 Jun 2026). Within this formulation, CGE is not merely inter-category separation but residual category geometry that survives controls for affect, style, topic, and coarse domain membership.

6. Geometric objects, symbolic structure, and cross-domain synthesis

The dissertation on geometric relational embeddings expands CGE beyond point clouds and text prototypes by embedding categories or concepts directly as geometric objects. The general setup defines entity embeddings

$S : \mathbb{R}^3 \to \mathbb{R}^M.$01

relation embeddings

$S : \mathbb{R}^3 \to \mathbb{R}^M.$02

and concept embeddings

$S : \mathbb{R}^3 \to \mathbb{R}^M.$03

where $S : \mathbb{R}^3 \to \mathbb{R}^M.$04 may consist of boxes, balls, cones, or other regions in Euclidean, hyperbolic, spherical, or pseudo-Riemannian spaces (Xiong, 2024).

In BoxEL, concepts are axis-aligned boxes

$S : \mathbb{R}^3 \to \mathbb{R}^M.$05

individuals are points, and roles are affine transformations

$S : \mathbb{R}^3 \to \mathbb{R}^M.$06

Subsumption $S : \mathbb{R}^3 \to \mathbb{R}^M.$07 is modeled by box containment and trained with

$S : \mathbb{R}^3 \to \mathbb{R}^M.$08

while conjunction, existential restriction, and disjointness are handled by corresponding volume-based losses (Xiong, 2024). In HMI, labels are represented by Poincaré hyperplanes and enclosing Euclidean balls, implication is represented by ball insideness, and exclusion by ball disjointness. In UltraE and QGCN, mixed-curvature or pseudo-Riemannian manifolds are used to capture both hierarchical and cyclic structure. In ShrinkE and FactE, high-order structures over qualifiers and nested facts are encoded geometrically (Xiong, 2024).

These formulations differ substantially from the pointwise semantic fields of GenDP, the canonical shape family of GCE-Pose, and the Wasserstein geometry tensor of category-level geometry learning segmentation. Nevertheless, they share a common principle: category structure is expressed through geometry itself. In GenDP, the shared structure is part-level 3D activation over object categories; in GCE-Pose, it is a category prototype plus deformation space; in domain generalization for segmentation, it is a set of class-specific geometric bases and a correlation matrix; in the LLM study, it is prototype geometry compared to an expert RDM; in geometric relational embeddings, it is explicit region geometry with logical relations (Wang et al., 2024, Li et al., 6 Feb 2025, He et al., 15 Aug 2025, Zhu et al., 22 Jun 2026, Xiong, 2024).

Several limitations recur across these settings. GenDP explicitly notes fixed reference features per task and points to long-horizon, multi-stage tasks as a challenge, suggesting task-adaptive semantic fields and more explicit geometric modeling such as part-based kinematic structures or SE(3)-equivariant embeddings (Wang et al., 2024). GCE-Pose assumes a known category and category-specific prototypes, and its global context prior is learned within that category (Li et al., 6 Feb 2025). The category-level geometry learning segmentation framework depends on labeled source data, class-wise geometry construction, and Sinkhorn-based optimal transport, and notes issues of scalability, class imbalance, and computational overhead (He et al., 15 Aug 2025). The LLM study states that its expert matrix is a structured external benchmark for category-level comparison rather than a clinical diagnostic ontology, and that zero-shot alignment remains moderate (Zhu et al., 22 Jun 2026). The dissertation highlights the cost of non-Euclidean operations and the limits of fixed geometric inductive biases (Xiong, 2024).

Taken together, these works support a technically specific view of CGE. It is not a single architecture and not a single metric. Rather, it is a family of representations in which categories are given geometric form through semantic fields, canonical prototypes, deformation spaces, per-class geometry bases, prototype RDMs, or geometric regions; and in which downstream performance depends on whether that geometry preserves the category structure relevant to the task (Wang et al., 2024, Li et al., 6 Feb 2025, He et al., 15 Aug 2025, Zhu et al., 22 Jun 2026, Xiong, 2024).