Any-dimensional equivariant neural networks
Eitan Levin Mateo Díaz
Traditional supervised learning aims to learn an unknown mapping by fitting a
function to a set of input-output pairs with a fixed dimension. The fitted
function is then defined on inputs of the same dimension. However, in many
settings, the unknown mapping takes inputs in any dimension; examples include
graph parameters defined on graphs of any size and physics quantities defined
on an arbitrary number of particles. We leverage a newly-discovered phenomenon
in algebraic topology, called representation stability, to define equivariant
neural networks that can be trained with data in a fixed dimension and then
extended to accept inputs in any dimension. Our approach is user-friendly,
requiring only the network architecture and the groups for equivariance, and
can be combined with any training procedure. We provide a simple open-source
implementation of our methods and offer preliminary numerical experiments.