Permutation Equivariant Neural Networks for Symmetric Tensors
This work addresses the need for models that respect symmetries in data for applications in physics, chemistry, and materials science, but it is incremental as it builds on existing permutation equivariance research by focusing on symmetric tensors.
The paper tackled the problem of designing permutation equivariant neural networks for symmetric tensors, which are common in various scientific fields, by characterizing all linear permutation equivariant functions between symmetric power spaces. The result showed that these functions are highly data efficient compared to standard MLPs and have potential for generalization to different tensor sizes.
Incorporating permutation equivariance into neural networks has proven to be useful in ensuring that models respect symmetries that exist in data. Symmetric tensors, which naturally appear in statistics, machine learning, and graph theory, are essential for many applications in physics, chemistry, and materials science, amongst others. However, existing research on permutation equivariant models has not explored symmetric tensors as inputs, and most prior work on learning from these tensors has focused on equivariance to Euclidean groups. In this paper, we present two different characterisations of all linear permutation equivariant functions between symmetric power spaces of $\mathbb{R}^n$. We show on two tasks that these functions are highly data efficient compared to standard MLPs and have potential to generalise well to symmetric tensors of different sizes.