Permutation Equivariant Neural Networks for Symmetric Tensors (2503.11276v2)

Abstract: 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.

Analysis of Permutation Equivariant Neural Networks for Symmetric Tensors

The paper "Permutation Equivariant Neural Networks for Symmetric Tensors" by Edward Pearce-Crump introduces a novel approach to incorporating permutation equivariance into neural networks with symmetric tensors as inputs. The notion of symmetry is pivotal, particularly in data structures such as symmetric tensors, which emerge in diverse areas like statistics, graph theory, physics, and materials science. This research seeks to address the limitations of previous models, which predominantly centered on equivariance to Euclidean groups and did not leverage symmetric tensors in permutation equivariant models.

Key Contributions

The paper's main contributions can be distilled into several critical aspects:

1. Characterization of Linear Permutation Equivariant Functions: Two distinct formulations of linear permutation equivariant functions between symmetric power spaces of $R^n$ are provided. This work extends our understanding beyond the Euclidean-focused approaches in prior research.
2. Introduction of Map Label Notation: To efficiently address the practical challenges in implementation associated with storing large weight matrices in memory, the paper introduces map label notation. This innovative approach enables the transformation of symmetric tensors by permutation equivariant weight matrices without requiring explicit storage, thus optimizing memory usage and computational efficiency.
3. Empirical Validation: Through the validation on toy problems, the paper demonstrates that permutation equivariant neural networks exhibit high data efficiency compared to standard MLPs. Furthermore, these networks showcase potential for generalization across symmetric tensors of varying sizes.

Numerical Results

In the synthetic permutation invariant task evaluated with symmetric tensors, the proposed model demonstrated superior data efficiency when juxtaposed with conventional MLPs as illustrated by a reduced test MSE (0.0447 for PermEquiv against 0.6486 for MLP). Furthermore, in the permutation equivariant task aimed at extracting diagonals from tensors, the model not only achieved lower test errors (SymmPermEquiv: 0.0035) compared to an equivariant model (0.0447) and MLP (0.6486) but also displayed capability in generalizing well across tensors of different sizes.

Implications and Future Directions

The impact of successfully characterizing permutation equivariant functions for symmetric tensors is multi-faceted. The efficient characterization and the memory-optimized transformation method could significantly enhance applications in fields like fluid dynamics, materials science, and neuroscience—where symmetric tensor data structures are prevalent.

In terms of theoretical implications, this research adds to the foundational knowledge of symmetric tensor manipulation within neural network architectures. The shift from Euclidean group equivariance to permutation-focused symmetry paradigms marks a progression that can inspire further exploration of symmetry-aware models.

Looking into the future, the potential for advancing symmetrical tensor comprehension in AI could be expanded by investigating unsupervised and reinforcement learning environments where permutation symmetries naturally occur. Additionally, there lies scope to integrate these models in real-world applications needing symmetry invariance such as drug discovery and complex system simulations.

By embedding symmetric tensors into wider tensor spaces and employing permutation equivariant frameworks, this paper paves the way for deeper integration of symmetry principles in AI architectures, offering a robust pathway towards model efficiency and generalization improvement.