All models discussed in the paper use ReLU non-linearity.

It is common that these data themselves have inherently complicated 3D geometric structures.

It is common that these data themselves have inherently complicated 3D geometric structures.

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

The ability to perform fast and accurate atomistic simulations is crucial for advancing the chemical sciences. By learning from high-quality data, machine-learned interatomic potentials achieve accuracy on par with ab initio and first-principles methods at a fraction of their computational cost. The success of machine-learned interatomic potentials arises from integrating inductive biases such as equivariance to group actions on an atomic system, e.g., equivariance to rotations and reflections. In particular, the field has notably advanced with the emergence of equivariant message passing. Most of these models represent an atomic system using spherical tensors, tensor products of which require complicated numerical coefficients and can be computationally demanding. Cartesian tensors offer a promising alternative, though state-of-the-art methods lack flexibility in message-passing mechanisms, restricting their architectures and expressive power. This work explores higher-rank irreducible Cartesian tensors to address these limitations. We integrate irreducible Cartesian tensor products into message-passing neural networks and prove the equivariance and traceless property of the resulting layers. Through empirical evaluations on various benchmark data sets, we consistently observe on-par or better performance than that of state-of-the-art spherical and Cartesian models.

We revisit the classic randomized sketch of a tensor product of $q$ vectors $x_i\in\mathbb{R}^n$.