Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.

翻译：张量是时间序列分析、材料科学及物理学等诸多科学领域中的一种基础数据结构。提升我们生成与处理张量的能力，对于高效解决这些领域中的问题至关重要。本文阐述了如何利用张量到张量映射函数所蕴含的底层对称性。具体而言，我们构建了在张量上具有普遍表达能力的等变机器学习架构，该架构利用了以下事实：在许多情况下，这类张量函数对于正交群、洛伦兹群和/或辛群的对角作用具有等变性。我们通过材料科学、理论计算机科学和时间序列分析中的三个问题展示了研究成果。针对时间序列，我们将本方法与日益流行的路径签名方法相结合，后者同样对重参数化具有不变性。数值实验表明，我们的等变模型性能优于对应的非等变基线模型。