Any-dimensional equivariant neural networks
This addresses a fundamental limitation in supervised learning for applications like graph analysis and particle physics, enabling models to handle variable input sizes without retraining.
The authors tackled the problem of learning mappings that accept inputs of any dimension, such as graph parameters or physics quantities, by developing equivariant neural networks that can be trained on fixed-dimension data and extended to any dimension using representation stability from algebraic topology. They provided a user-friendly implementation and preliminary experiments.
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.