Equivariant Frames and the Impossibility of Continuous Canonicalization
Nadav Dym Hannah Lawrence Jonathan W. Siegel
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.