Equivariant Frames and the Impossibility of Continuous Canonicalization
This solves a fundamental robustness issue in equivariant machine learning, impacting fields like molecular modeling and computer vision where continuous symmetry is crucial.
The paper tackles the problem of unweighted frame-averaging causing discontinuities in equivariant functions, showing that for groups like SO(2), SO(3), and S_n, no efficiently computable continuous canonicalization exists. It addresses this by formally defining and constructing weighted frames that provably preserve continuity, enabling efficient and continuous equivariant methods.
Canonicalization provides an architecture-agnostic method for enforcing equivariance, with generalizations such as frame-averaging recently gaining prominence as a lightweight and flexible alternative to equivariant architectures. Recent works have found an empirical benefit to using probabilistic frames instead, which learn weighted distributions over group elements. In this work, we provide strong theoretical justification for this phenomenon: for commonly-used groups, there is no efficiently computable choice of frame that preserves continuity of the function being averaged. In other words, unweighted frame-averaging can turn a smooth, non-symmetric function into a discontinuous, symmetric function. To address this fundamental robustness problem, we formally define and construct \emph{weighted} frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of $SO(2)$, $SO(3)$, and $S_n$ on point clouds.