拟阵约束下投影最大化的子空间选择
Subspace selection for projection maximization with matroid constraints
摘要：
假设在希尔伯特空间中存在由大量向量组成的基集。考虑选择基集的子集的问题，使得感兴趣的向量在所选子集中的向量所张成的子空间上的投影达到最大范数。这个问题通常是NP难的，并且已经提出了替代的近似算法，例如前向回归和正交匹配追踪作为启发式方法。在本文中，我们调查这些算法的性能上的界限，通过引入元素曲率的概念。更具体地说，我们推导出下界，作为这些元素的曲率的函数，上述算法的性能相对于均匀和非均匀拟阵约束下的最优解，分别。我们表明，如果在地面设置的元素是相互正交的，那么这些算法是最佳的拟阵时是均匀的，他们至少实现了$1/2$ -近似的最优解时的拟阵是不均匀的。
英文摘要：
Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by the vectors in the chosen subset reaches the maximum norm. This problem is generally NP-hard, and alternative approximation algorithms such as forward regression and orthogonal matching pursuit have been proposed as heuristic approaches. In this paper, we investigate bounds on the performance of these algorithms by introducing the notions of elemental curvatures. More specifically, we derive lower bounds, as functions of these elemental curvatures, for performance of the aforementioned algorithms with respect to that of the optimal solution under uniform and non-uniform matroid constraints, respectively. We show that if the elements in the ground set are mutually orthogonal, then these algorithms are optimal when the matroid is uniform and they achieve at least $1/2$-approximations of the optimal solution when the matroid is non-uniform.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
