The introduction of convolution and attention to the space of rays in 3D required additional geometric representations for which there was no space in the main paper to elaborate. We will introduce here all the necessary notations and definitions. We have accompanied this presentation with examples of specific groups to elucidate the abstract concepts needed in the definitions. Figure 10: The visualization of Plücker coordinates: A ray xcan be denoted as (d,m)where x is any point on the ray x, and dis the direction of the ray x. mis defined as x d. Given the action of the group G on a homogeneous space X, and given x0 as the origin of X, the stabilizer group H of x0 in G is the group that leaves x0 intact, i.e., H = {h G|hx0 = x0}. The group, G, can be partitioned into the quotient space (the set of left cosets) G/H and X is isomorphic to G/H since all group elements in the same coset transform x0 to the same element in X, that is, for any element g gH we have g x0 = gx0. Example 1. SE(3) acting on the ray space R: Take SE(3) as the acting group and the ray space R as its homogeneous space. We use Plücker coordinates to parameterize the ray space R: any x R can be denoted as (d,m), where d S2 is the direction of the ray, and m = x d where x is any point on the ray, as shown in figure 10. R is the quotient space SE(3)/(SO(2) R)up to isomorphism. Example 2. SE(3) acting on the 3DEuclidean space R3: R3 is isomorphic to SE(3)/SO(3). Consider another case when SE(3) acts on the homogeneous space R3; for any g = (R,t) SE(3) and x R3, gx = Rx+t. If the fixed origin is [0,0,0]T, the stabilizer subgroup is H = SO(3) since any rotation g = (R,0)leaves [0,0,0]T unchanged. The last example is SO(3) acting on the homogeneous space sphere S2. Given the fixed origin point as [0,0,1]T, the stabilizer group is SO(2).

We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as drop in replacement for non-equivariant convolutions.

Group theory has been used in machine learning to provide a theoretically grounded approach for incorporating known symmetry transformations in tasks from robotics to protein modeling. In these applications, equivariant neural networks use knownsymmetry groups with predefined representations to learn over geometric input data. We propose MatrixNet, a neural network architecture that learns matrix representations of group element inputs instead of using predefined representations. MatrixNet achieves higher sample efficiency and generalization over several standard baselines in prediction tasks over the several finite groups and the Artin braid group. We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.

We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs $(\mathbf{y}, g \cdot \mathbf{y})$, where $g$ is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps -- without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group $G:= (R_m \times \mathbb{Z}_n \times \mathbb{Z}_n)$, combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group $O(n)$ and the general linear group $GL(n)$. We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.

We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.