Recent work has shown the utility of developing machine learning models that respect the structure and symmetries of eigenvectors. These works promote sign invariance, since for any eigenvector v the negation -v is also an eigenvector. However, we show that sign invariance is theoretically limited for tasks such as building orthogonally equivariant models and learning node positional encodings for link prediction in graphs. In this work, we demonstrate the benefits of sign equivariance for these tasks. To obtain these benefits, we develop novel sign equivariant neural network architectures. Our models are based on a new analytic characterization of sign equivariant polynomials and thus inherit provable expressiveness properties. Controlled synthetic experiments show that our networks can achieve the theoretically predicted benefits of sign equivariant models. Code is available at https://github.com/cptq/Sign-Equivariant-Nets.

翻译：近期工作表明，开发尊重特征向量结构和对称性的机器学习模型具有实用性。这些研究倡导符号不变性，因为对于任何特征向量v，其负向量-v也是特征向量。然而，我们证明符号不变性在构建正交等变模型、学习图链接预测的节点位置编码等任务中存在理论局限性。本研究展示了符号等变性在这些任务中的优势。为获取这些优势，我们开发了新颖的符号等变神经网络架构。该模型基于符号等变多项式的全新解析特征刻画，因此继承了可证明的表达能力。受控合成实验表明，我们的网络能够实现符号等变模型的理论预测优势。代码见 https://github.com/cptq/Sign-Equivariant-Nets。