# Equivariant Maps for Hierarchical Structures

arXiv: https://arxiv.org/abs/2006.03627 · HTML: https://mlsanity.com/paper/2006.03627/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2020.

## Built on by (most cited first, 11 of 11)

- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [SQN: Weakly-Supervised Semantic Segmentation of Large-Scale 3D Point Clouds](https://mlsanity.com/paper/2104.04891/) (2021)
- [Equivariant Deep Weight Space Alignment](https://mlsanity.com/paper/2310.13397/) (2023)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [On the Expressive Power of Spectral Invariant Graph Neural Networks](https://mlsanity.com/paper/2406.04336/) (2024)
- [Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations](https://mlsanity.com/paper/2410.06665/) (2024)
- [Augmenting Imitation Experience via Equivariant Representations](https://mlsanity.com/paper/2110.07668/) (2021)

## Abstract (arXiv)

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.