There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars---scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space.We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties.

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality d . The key observation is that nonlinear O( d)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars---scalar products and scalar contractions of the scalar, vector, and tensor inputs.