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Equivariant Frames and the Impossibility of Continuous Canonicalization

Nadav Dym, Hannah Lawrence, Jonathan W. Siegel

TL;DR

The paper proves fundamental limits on continuous canonicalization for key group actions ($S_n$, $SO(d)$, and $O(d)$) and shows that unweighted frame-averaging often induces discontinuities. To restore robustness, it introduces weighted, robust frames that are continuous and (weakly) equivariant, and constructs practical instances for permutations and rotations, including explicit cardinality bounds. These robust frames yield invariant (and, with stable backbones, equivariant) projection operators that preserve continuity, enabling universal, continuous models for group actions on point clouds. The work connects with probabilistic/weighted-frame approaches in the literature and demonstrates, both theoretically and experimentally, that weighted frames can outperform discontinuous canonicalization while remaining computationally feasible. This provides a principled, scalable path to robust, continuous equivariant learning across common geometric groups.

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.

Equivariant Frames and the Impossibility of Continuous Canonicalization

TL;DR

The paper proves fundamental limits on continuous canonicalization for key group actions (, , and ) and shows that unweighted frame-averaging often induces discontinuities. To restore robustness, it introduces weighted, robust frames that are continuous and (weakly) equivariant, and constructs practical instances for permutations and rotations, including explicit cardinality bounds. These robust frames yield invariant (and, with stable backbones, equivariant) projection operators that preserve continuity, enabling universal, continuous models for group actions on point clouds. The work connects with probabilistic/weighted-frame approaches in the literature and demonstrates, both theoretically and experimentally, that weighted frames can outperform discontinuous canonicalization while remaining computationally feasible. This provides a principled, scalable path to robust, continuous equivariant learning across common geometric groups.

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 , , and on point clouds.
Paper Structure (32 sections, 35 theorems, 189 equations, 2 figures, 3 tables)

This paper contains 32 sections, 35 theorems, 189 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Paper structure
  4. Projections, canonicalization, and frames
  5. Preliminaries
  6. Projection operators
  7. Canonicalization
  8. Canonicalization-based equivariant projection operators.
  9. Examples where there is no continuous canonicalization.
  10. Frames
  11. Weighted frames
  12. Continuous weighted frames
  13. Weighted frames for permutations
  14. Robust frames for permutations on $\mathbb{R}^{d\times n}_{\mathrm{distinct}}$
  15. Robust frames for all of $\mathbb{R}^{d\times n}$
  16. ...and 17 more sections

Key Result

Proposition 2.1

Let $\mathcal{E}:F(V,W)\to F(V,W)$ be a BEC operator, and $Q\subseteq C(V,W)$ a dense subset . Then $\mathcal{E}(Q)=\{\mathcal{E}(q)| ~ q \in Q\}$ contains only continuous equivariant functions, and is dense in $C_{\mathrm{equi}}(V,W)$ .

Figures (2)

  • Figure 1: Left: Canonicalization and its generalization to frame averaging. Under group transformation of the input, both its canonicalization and the set of inputs transformed by the frame are invariant. Right: Group averaging and group canonicalization are special cases of frames of maximal and minimal size, respectively. Frames, in turn, are a special case of weighted frames.
  • Figure 2: Left: Examples of natural canonicalizations. Canonicalizing $\mathbb{R}^n$ with respect to $S_n$ via sorting is continuous, but the lexicographic generalization to $\mathbb{R}^{2 \times n}$ is not. Similarly, one can canonicalize $2D$ ordered point clouds with respect to $SO(2)$ by applying the rotation that aligns the first node with the positive $x$-axis, but this is discontinuous. Right: Visualization of a weakly equivariant frame $\mu$ for $SO(2)$ acting on an unordered point cloud $v$ with $120^\circ$ self-symmetry, evaluated both at $v$ and at $gv$, a $60^\circ$ rotation of $v$. $\mu_{gv}$ is not a $60^\circ$ rotation of $\mu_v$, but $\overline{\mu}$ is equivariant by definition, so $\overline{\mu}_{gv}$is a $60^\circ$ rotation of $\overline{\mu}_v$. Thanks to the self-symmetry of $v$, $v$ is exactly the same whether rotated by $0^\circ$, $120^\circ$, or $240^\circ$. Thus, invariant projection by $\mu$ or $\overline{\mu}$ has the same result; $\mu$ is simply $3x$ more efficient.

Theorems & Definitions (71)

  • Definition 2.1: BEC operator
  • Proposition 2.1
  • Definition 2.2: Orbit canonicalization
  • Example 2.3
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.4
  • proof : Proof idea
  • Theorem 2.5
  • ...and 61 more