Equivariant Frames and the Impossibility of Continuous Canonicalization
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916292636704768 |
|---|---|
| author | Dym, Nadav Lawrence, Hannah Siegel, Jonathan W. |
| author_facet | Dym, Nadav Lawrence, Hannah Siegel, Jonathan W. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16077 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Frames and the Impossibility of Continuous Canonicalization Dym, Nadav Lawrence, Hannah Siegel, Jonathan W. Machine Learning 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. |
| title | Equivariant Frames and the Impossibility of Continuous Canonicalization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2402.16077 |