Rotation Equivariant Operators for Machine Learning on Scalar and Vector Fields

TL;DR

This paper develops a comprehensive theory and software for rotation equivariant operators on scalar and vector fields, enabling symmetry-preserving computations in simulation, machine learning, and inverse problems.

Contribution

It extends convolution theorems to characterize linear equivariant operators via tensor field convolutions and provides a Julia package for practical implementation.

Findings

Operators are characterized by tensor convolutions with radially symmetric kernels.

Most Green's functions and differential operators are inherently equivariant.

The software enables simulation, image processing, and inverse problem solving with equivariance.

Abstract

We develop theory and software for rotation equivariant operators on scalar and vector fields, with diverse applications in simulation, optimization and machine learning. Rotation equivariance (covariance) means all fields in the system rotate together, implying spatially invariant dynamics that preserve symmetry. Extending the convolution theorems of linear time invariant systems, we theorize that linear equivariant operators are characterized by tensor field convolutions using an appropriate product between the input field and a radially symmetric kernel field. Most Green's functions and differential operators are in fact equivariant operators, which can also fit unknown symmetry preserving dynamics by parameterizing the radial function. We implement the Julia package EquivariantOperators.jl for fully differentiable finite difference equivariant operators on scalar, vector and higher order tensor fields in 2d/3d. It can run forwards for simulation or image processing, or be back propagated for computer vision, inverse problems and optimal control. Code at https://aced-differentiate.github.io/EquivariantOperators.jl/