PolyhedronNet: Representation Learning for Polyhedra with Surface-attributed Graph

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 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. In mathematics and computational geometry Yu et al. (2025), a polyhedron is defined as a threedimensional (3D) solid formed by flat polygon faces joined at edges and vertices. Ubiquitous geometric shapes can be precisely and efficiently modeled as polyhedra, ranging from basic 3D shapes (e.g., cubic, pyramid, and truncated tetrahedron) to compositions of them (e.g., shapes of buildings, furniture, and digital objects in CAD) as exemplified Figure 1 (a). In the real world, there are many tasks surrounding polyhedra such as classification (e.g., convex or concave); clustering polyhedra into different types (e.g., Platonic solids and prisms); as well as generation and optimization (e.g., use faceted facades to break up flat surfaces) of polyhedra for design needs.