What are some interesting functions that are equivariant under rotations in SO(3)? 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).
 A: 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 $SO(3)$.) 
A: 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
A: 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.
