论文标题
对谎言组的各种优化，并提供了领先（广义）特征值问题的示例
Variational Optimization on Lie Groups, with Examples of Leading (Generalized) Eigenvalue Problems
论文作者
论文摘要
本文考虑了谎言组功能的平稳优化。通过将矢量空间中的nag变异原理概括为谎言基团，可以确保获得连续的lie-nag动力学，这些动态可以得到融合到局部最佳最佳。它们对应于谎言组上梯度流动的动量版本。然后对$ \ Mathsf {so}（n）$的特定情况进行详细研究，其目标功能与领先的广义特征值问题相对应：首先在坐标中明确地将lie-nag动态变为坐标，然后在结构中离散，然后在结构中置换，从而保留了时尚，从而在忠实的能量上置于忠实的能量的层次（应当构成综合性），并确切地构成了综合性（应当构成综合性）。还研究了随机梯度版本。关于合成数据和实际问题的数值实验（MNIST的LDA）证明了所提出方法作为优化算法的有效性（$而不是$作为分类方法）。
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).
