Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems. Given a group acting on the input space, one augments the training set with transformed copies of each sample. Because it exploits symmetries without modifying the underlying learning algorithm, data augmentation can be applied broadly across learning methods. However, this universality comes at a computational cost: when the group is large, full group-sized augmentation quickly becomes computationally infeasible. This raises a fundamental question: Can partial data augmentation achieve the same statistical benefits as full augmentation in terms of generalization and sample complexity? We develop a general framework for investigating this question using Fourier analysis and the representation theory of finite groups. We show that, for a broad class of classical learning problems, partial data augmentation based on a randomly sampled subset of group elements achieves the same minimax rates as full augmentation, up to an approximation error that vanishes as the subset size increases. Our results provide a theoretical explanation for why partial augmentation can retain the statistical benefits of full augmentation despite enforcing symmetry only approximately, and shed light on a recently raised question in learning with symmetries: whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods. Moreover, we prove a complementary impossibility result: enforcing exact invariance via data augmentation requires averaging over the entire group, and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive. Together, these results provide a unified perspective on full and partial data augmentation, as well as exact and approximate symmetry enforcement.

Equivariant Graph Neural Networks (EGNNs) have demonstrated significant success in modeling microscale systems, including those in chemistry, biology and materials science.

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce TensorNet, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.

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).

Convolutional Neural Networks (CNNs) (LeCun et al., 1989) have become a widely used and powerful tool for computer vision tasks, in large part due to their ability to achieve translation