论文标题
在最小封闭的线性子空间中
On a minimum enclosing ball of a collection of linear subspaces
论文作者
论文摘要
本文涉及一系列线性子空间集合的最小值中心。当子空间为$ k $二维子空间$ \ mathbb {r}^n $时，可以将其施放为在Grassmann歧管上找到最小封闭球的中心，Gr $（k，n）$。对于不同维度的子空间，设置变成了司法的不一致的结合，而不是单个歧管，问题不再定义。但是，这些流形之间存在自然的几何图，对于映射下的子空间的图像，距离有明确的距离概念。在此上下文中解决初始问题会导致每个组成歧管上的候选最小值中心，但并未固有地提供有关哪个候选者是数据最佳表示的直觉。此外，通常不嵌套不同等级的解决方案，因此通缩方法不足以足够，并且必须在每个歧管上独立解决问题。我们提出并解决了由Minimax中心等级参数参数的优化问题。使用双重级别算法计算该解决方案。通过扩展目标并惩罚排名-K $ minimax中心丢失的信息，我们共同恢复最佳尺寸，$ k^*$和一个中央子空间，$ u^*\ in $ gr $（k^*，n）$在最小封闭球的中心，最小代表数据。
This paper concerns the minimax center of a collection of linear subspaces. When the subspaces are $k$-dimensional subspaces of $\mathbb{R}^n$, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold, Gr$(k,n)$. For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose and solve an optimization problem parametrized by the rank of the minimax center. The solution is computed using a subgradient algorithm on the dual. By scaling the objective and penalizing the information lost by the rank-$k$ minimax center, we jointly recover an optimal dimension, $k^*$, and a central subspace, $U^* \in$ Gr$(k^*,n)$ at the center of the minimum enclosing ball, that best represents the data.
