  We consider achieving equivariance in machine learning systems via frame
averaging. Current frame averaging methods involve a costly sum over large
frames or rely on sampling-based approaches that only yield approximate
equivariance. Here, we propose Minimal Frame Averaging (MFA), a mathematical
framework for constructing provably minimal frames that are exactly
equivariant. The general foundations of MFA also allow us to extend frame
averaging to more groups than previously considered, including the Lorentz
group for describing symmetries in space-time, and the unitary group for
complex-valued domains. Results demonstrate the efficiency and effectiveness of
encoding symmetries via MFA across a diverse range of tasks, including $n$-body
simulation, top tagging in collider physics, and relaxed energy prediction. Our
code is available at https://github.com/divelab/MFA.
