李群上的变分最优化，以前导（广义）特征值问题为例
Variational Optimization on Lie Groups, with Examples of Leading (Generalized) Eigenvalue Problems
摘要：
本文考虑李群上函数的光滑优化。通过在向量空间中推广NAG变分原理（Wibisono等人，2016）应用于李群，得到了保证收敛于局部最优的连续Lie-NAG动力学.它们对应于李群上梯度流的动量形式。一个特殊的情况下，$\mathsf{SO}(n)$，然后详细研究，与目标函数对应的领先的广义特征值问题：李NAG动力学的第一次明确的坐标，然后离散化的结构保持的方式，从而在优化算法与忠实的能量行为（由于共形辛性）和完全保持在李群。随机梯度版本也进行了研究。数值实验的合成数据和实际问题（LDA MNIST）证明了所提出的方法作为优化算法（$not$作为分类方法）的有效性。
英文摘要：
The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of $\mathsf{SO}(n)$ is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms ($not$ as a classification method).
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