Abstract: Lie group not only has algebraic structures, but also has manifold structures. Data are mapped to multiply Lie group, according to the track ralation of Lie group samples on the manifold, these tracks which are homotopic can be covered. So, these covering areas can present the category information. This paper proposes a new covering algorithm called multiply Lie group kernel covering learning algorithm that uses kernel function to make the covering area have more divisibility and the edge of covering area smooth. The experimental results on the MNIST datasets show that the proposed algorithm has better classification performance with comparison to the other algorithm called multiply connected Lie group covering learning algorithm.

Key words:  Lie group, manifold, covering learning algorithm, kernel function

摘要： 李群具有代数结构也具有流形几何结构。将数据映射到多李群空间，并根据李群样本点在李群流形上的轨道关系，对那些同伦的轨道加以覆盖，从而使得覆盖域呈现出类别信息。利用核函数的思想，进一步使得类别不同的覆盖域更具有可分性，同时覆盖边界更具有光滑性，因此提出了多李群核覆盖学习算法。在MNIST手写体数字图像上进行了多组实验验证，并对实验结果进行了分析，结果表明与多连通李群覆盖学习算法相比，多李群核覆盖学习算法具有较好的分类效果。

关键词: 李群, 流形结构, 覆盖学习算法, 核函数