Equivariant Frames and the Impossibility of Continuous Canonicalization

Nadav Dym, Hannah Lawrence, Jonathan W. Siegel

Research output: Contribution to journalConference articlepeer-review

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 weighted frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of SO(d), O(d), and Sn on point clouds.

Original languageEnglish
Pages (from-to)12228-12267
Number of pages40
JournalProceedings of Machine Learning Research
Volume235
StatePublished - 2024
Externally publishedYes
Event41st International Conference on Machine Learning, ICML 2024 - Vienna, Austria
Duration: 21 Jul 202427 Jul 2024

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

Cite this