Equivariant Maps for Hierarchical Structures
This provides a foundational framework for modeling hierarchical structures in machine learning, addressing a gap in handling complex real-world data like point clouds.
The paper tackles the lack of a formalism for applying deep learning to hierarchical data structures, such as sequences of sets or graphs of graphs, by developing equivariant maps based on wreath product symmetries. It demonstrates effectiveness by achieving state-of-the-art results on Semantic3D, S3DIS, and vKITTI point-cloud segmentation benchmarks.
While using invariant and equivariant maps, it is possible to apply deep learning to a range of primitive data structures, a formalism for dealing with hierarchy is lacking. This is a significant issue because many practical structures are hierarchies of simple building blocks; some examples include sequences of sets, graphs of graphs, or multiresolution images. Observing that the symmetry of a hierarchical structure is the "wreath product" of symmetries of the building blocks, we express the equivariant map for the hierarchy using an intuitive combination of the equivariant linear layers of the building blocks. More generally, we show that any equivariant map for the hierarchy has this form. To demonstrate the effectiveness of this approach to model design, we consider its application in the semantic segmentation of point-cloud data. By voxelizing the point cloud, we impose a hierarchy of translation and permutation symmetries on the data and report state-of-the-art on Semantic3D, S3DIS, and vKITTI, that include some of the largest real-world point-cloud benchmarks.