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I'm interested in machine learning on 3D point clouds. Are there any interesting functions that are equivariant under rotations in SO(3)?

The PointNet paper: https://arxiv.org/abs/1612.00593

already found one way to design functions that are invariant to permutations of the inputs (symmetric), but I haven't found a class of functions that is both symmetric and also equivariant under SO(3).

The functions don't have to be complex themselves as long as they can be composed with each other to approximate any continuous function that is symmetric and equivariant under SO(3).

Andrew Wagner
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  • what do you mean by equivariant? something like $f(R.x)=R.f(x), R\in\mathrm{SO}(3)$? or just $f(R.x)=f(x), R\in\mathrm{SO}(3)$? If it is the former case, how to you define the action on the function? It is vector function? – Troy Woo Nov 23 '17 at 15:40
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    Yep, the first one is the "equivariance" I'm asking about, the second one is invariance. For this to make sense f has to be R^3->R^3. – Andrew Wagner Nov 23 '17 at 15:59
  • The first thing that comes to mind is of course momentum and angular momentum. I doubt that would be useful. – Troy Woo Nov 23 '17 at 16:07
  • Functions that act radially (i.e. on the length of the vector) are equivariant under rotation; that's used in some recent (2D equivariance) CNN papers: https://arxiv.org/pdf/1612.09346.pdf. That feels trivial, but maybe that's as rich as it gets... – Andrew Wagner Nov 23 '17 at 16:19
  • I have no idea. Does RBF help? – Troy Woo Nov 23 '17 at 16:51
  • "Radial Function" is indeed the wikipedia-standard term. Radial Basis Function also turns up a bunch of relevant stuff for me to sift through. Thanks! – Andrew Wagner Nov 24 '17 at 08:38
  • You might want to consider accepting Yuan's answer. For my clumsy mind, his answer misses one step, namely to show that $f(x)$ is always parallel to $x$, which amounts to showing both are eigenvectors of a particular rotation matrix. – Troy Woo Nov 24 '17 at 09:50

3 Answers3

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If $g : \mathbb{R} \to \mathbb{R}$ is a function, then the function

$$f : \mathbb{R}^3 \ni v \mapsto g(|v|) v \in \mathbb{R}^3$$

is $SO(3)$-equivariant. This function is continuous iff $g$ is continuous and smooth iff $g(x)$ is a smooth function of $x^2$.

This exhausts all possibilities, for the following reason. If $f : \mathbb{R}^3 \to \mathbb{R}^3$ is $SO(3)$-equivariant then it must send orbits under the action of $SO(3)$ to orbits. These orbits are precisely the spheres around the origin, so $f$ must send the sphere of radius $r$ in $\mathbb{R}^3$ to the sphere of some radius $g(r)$.

Furthermore, there are exactly two $SO(3)$-equivariant functions from the sphere of radius $r$ to the sphere of radius $g(r)$, given by scaling by $\pm g(r)$: to see this note that every point $p$ on the sphere is almost uniquely specified by specifying the axis such that a rotation around that axis fixes $p$, up to antipodes. This is accounted for in the above construction by allowing $g$ to be negative.

(Aside: if $f$ is required to be a polynomial map then this question can be understood using representation theory. We recover the maps of the form $v \mapsto p(|v|^2) v$ where $p$ is a polynomial using a known decomposition of the symmetric algebra $\text{Sym}(\mathbb{R}^3)$ into irreducible representations of $SO(3)$.)

Qiaochu Yuan
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  • Cool. I didn't get it until I start to think about the eigenvector. – Troy Woo Nov 23 '17 at 20:15
  • Thanks! So among $f : \mathbb{R}^3 \to \mathbb{R}^3$ the only functions that are $SO(3)$-equivariant are radial functions. – Andrew Wagner Nov 24 '17 at 10:29
  • This is great! Just to finish the argument: let $p' = f(p)$, and consider any rotation $R_p$ that keeps $p$ fixed. Then, $p' = f(p) = f(R_p p) = R_p f(p) = R_p p'$. Thus, $R_p$ must keep $p'$ fixed as well. Thus, $f(p)$ and $p$ are parallel to each other. – Ameya Jul 08 '22 at 22:24
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The critical observation (from Yuan's answer) is that among $f : \mathbb{R}^3 \to \mathbb{R}^3$ the only functions that are $SO(3)$-equivariant are radial functions.

Wikipedia contains a list of examples of such functions:

https://en.wikipedia.org/wiki/Radial_basis_function

Andrew Wagner
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I realize this question was asked a while ago, but I wanted to chime in for completeness.

The spherical harmonics (the basis functions of the irreducible representations of $SO(3)$) are 3D-rotation equivariant. The coefficients of a signal represented by spherical harmonics can be rotated using the Wigner D matrix.

You can use these to construct rotation-equivariant convolutional neural networks as shown in the following papers: Tensor field networks*, 3D Steerable CNNs, and closely related Clebsch-Gordon nets.

*disclosure: I'm an author on this paper.

Tess
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