Equivariant Frames and the Impossibility of Continuous Canonicalization

TL;DR

This paper proves that unweighted frame-averaging cannot maintain continuity for certain groups, and introduces weighted frames that ensure continuity, improving the robustness of equivariant methods.

Contribution

It provides a theoretical foundation showing the limitations of unweighted frames and constructs continuous weighted frames for key symmetry groups.

Findings

Unweighted frame-averaging can cause discontinuities in symmetric functions.

Weighted frames can preserve continuity in equivariant functions.

Efficient continuous weighted frames are constructed for $SO(2)$, $SO(3)$, and $S_n$.

Abstract

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.