提出可处理对称张量的置换等变神经网络，提升数据效率与泛化能力。
Permutation Equivariant Neural Networks for Symmetric Tensors
- 基于对称幂空间构造线性置换等变函数，保持数据对称性
- 在两个任务上表现优于标准MLP，对不同尺寸张量泛化良好
- 适合物理、化学中涉及对称张量的数据建模场景
将置换等变性融入神经网络已被证明有助于模型尊重数据中的对称性。对称张量广泛存在于统计学、机器学习和图论中，在物理学、化学及材料科学等领域具有重要应用。然而，现有置换等变模型研究尚未涵盖对称张量作为输入的情况，且以往针对此类张量的学习工作多聚焦于欧几里得群等变性。本文给出了对称幂空间间所有线性置换等变函数的两种不同刻画。实验表明，这些函数在两项任务中相比标准MLP展现出更高的数据效率，并具备良好的泛化能力，能有效适应不同尺寸的对称张量。
原文摘要 · Abstract (English)
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.
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