用张量记号让矩阵李群运算更清晰，提升估计精度。
Improved Representation of Matrix Lie Group Operations through Tensor Notation
- 用张量和爱因斯坦求和约定表示矩阵李群操作
- 显著简化李群导数的推导与计算流程
- 适合需要精确推导的优化与估计研究者
近期多项研究证明，将李群应用于估计问题可提升准确性和一致性。本文提出一种新工具：用张量和爱因斯坦求和记号描述矩阵李群运算。尽管张量与爱因斯坦记号在其他领域已成熟，但将其用于矩阵李群导数的表示与计算仍属首次。更重要的是，该记号极大简化了梯度依赖估计框架中所需的操作与导数表达。因此，本文主要贡献并非新功能，而是一种更清晰的矩阵李群数学记号。
原文摘要 · Abstract (English)
Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.
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