Learning Hierarchical Hyperbolic Mixture Model for Part-aware 3D Generation

3D shape generation has become increasingly important for graphics and vision applications. Current part-aware 3D generation usually overlooks hierarchical part relations or inefficiently encodes multi-level semantics in Euclidean space. Thus we propose a novel framework for hierarchical and efficient part-aware 3D generation in hyperbolic space. Our contributions are three-fold: (1) Hierarchical Hyperbolic Mixture Model (H$^2$MM): We propose part-aware semantic representation of objects within a hyperbolic manifold, providing a high-fidelity hierarchical part-aware representation of object details and semantics. (2) Hyperbolic Semantically Consistent Diffusion Model: We design the geodesic diffusion process that preserves the hierarchical and semantic structure of H$^{2}$MM, and progressively generates semantics from conditions and generates object under their joint guidance. We use an adaptive tree-structured neural network to loosen the constraint of jointly generating nodes and edges in previous hyperbolic diffusion. (3) Hyperbolic Diffusion Model Solver: We leverage higher-order Riemannian gradient on hyperbolic manifolds for designing a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. Extensive experiments demonstrate that our method achieves superior quality and efficiency. Code will be public.