Article,
Equivariant Maps for Hierarchical Structures
(2020)cite arxiv:2006.03627.
Abstract
In many real-world settings, we are interested in learning invariant and
equivariant functions over nested or multiresolution structures, such as a set
of sequences, a graph of graphs, or a multiresolution image. While equivariant
linear maps and by extension multilayer perceptrons (MLPs) for many of the
individual basic structures are known, a formalism for dealing with a hierarchy
of symmetry transformations is lacking. Observing that the transformation group
for a nested structure corresponds to the ``wreath product'' of the symmetry
groups of the building blocks, we show how to obtain the equivariant map for
hierarchical data-structures using an intuitive combination of the equivariant
maps for the individual blocks. To demonstrate the effectiveness of this type
of model, we use a hierarchy of translation and permutation symmetries for
learning on point cloud data, and report state-of-the-art on semantic3d
and s3dis, two of the largest real-world benchmarks for 3D semantic
segmentation.
