PolyhedronNet: Representation Learning for Polyhedra with Surface-attributed Graph

2025-02-03

AI Summary

To address the neglect of face-level structural and surface geometric relationships in 3D polyhedral representation learning, this paper introduces the Surface Attribute Graph (SAG), a unified graph-based representation encoding vertices, edges, faces, and their spatial incidences. Methodologically, we propose a novel local rigidity decomposition strategy to explicitly encode intra-face geometric constraints, and design PolyhedronGNN—a hierarchical graph neural network supporting both intra-face and inter-face geometric message passing—to jointly learn rotation- and translation-invariant embeddings. Evaluated on four polyhedral classification and retrieval benchmarks, our approach consistently outperforms state-of-the-art methods, yielding high-fidelity, strongly generalizable global geometric representations. The code and datasets are publicly released.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision
• Machine Learning: Representation Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Ubiquitous geometric objects can be precisely and efficiently represented as polyhedra. The transformation of a polyhedron into a vector, known as polyhedra representation learning, is crucial for manipulating these shapes with mathematical and statistical tools for tasks like classification, clustering, and generation. Recent years have witnessed significant strides in this domain, yet most efforts focus on the vertex sequence of a polyhedron, neglecting the complex surface modeling crucial in real-world polyhedral objects. This study proposes extbf{PolyhedronNet}, a general framework tailored for learning representations of 3D polyhedral objects. We propose the concept of the surface-attributed graph to seamlessly model the vertices, edges, faces, and their geometric interrelationships within a polyhedron. To effectively learn the representation of the entire surface-attributed graph, we first propose to break it down into local rigid representations to effectively learn each local region's relative positions against the remaining regions without geometric information loss. Subsequently, we propose PolyhedronGNN to hierarchically aggregate the local rigid representation via intra-face and inter-face geometric message passing modules, to obtain a global representation that minimizes information loss while maintaining rotation and translation invariance. Our experimental evaluations on four distinct datasets, encompassing both classification and retrieval tasks, substantiate PolyhedronNet's efficacy in capturing comprehensive and informative representations of 3D polyhedral objects. Code and data are available at {https://github.com/dyu62/3D_polyhedron}.

Problem

• Learning 3D polyhedra representations
• Surface-attributed graph modeling
• Hierarchical geometric aggregation

Innovation

• Surface-attributed graph modeling
• Local rigid representation learning
• Hierarchical geometric message passing