Equivariant Maps for Hierarchical Structures

TL;DR

This paper introduces a method to construct equivariant maps for hierarchical data structures using the wreath product of symmetries, enabling deep learning models to better handle complex hierarchical data like point clouds.

Contribution

It formalizes the construction of equivariant maps for hierarchical structures and demonstrates their effectiveness in semantic segmentation of point-cloud data.

Findings

Achieved state-of-the-art results on Semantic3D, S3DIS, and vKITTI datasets.

Proposed a hierarchical equivariant map formalism based on wreath products.

Validated the approach on large real-world point-cloud benchmarks.

Abstract

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.