Given a set of points of interest, a volumetric spanner is a subset of the points using which all the points can be expressed using "small" coefficients (measured in an appropriate norm). Formally, given a set of vectors $X = \{v_1, v_2, \dots, v_n\}$, the goal is to find $T \subseteq [n]$ such that every $v \in X$ can be expressed as $\sum_{i\in T} \alpha_i v_i$, with $\|\alpha\|$ being small. This notion, which has also been referred to as a well-conditioned basis, has found several applications, including bandit linear optimization, determinant maximization, and matrix low rank approximation. In this paper, we give almost optimal bounds on the size of volumetric spanners for all $\ell_p$ norms, and show that they can be constructed using a simple local search procedure. We then show the applications of our result to other tasks and in particular the problem of finding coresets for the Minimum Volume Enclosing Ellipsoid (MVEE) problem.

翻译：给定一组兴趣点，体积极张量是这些点的一个子集，利用该子集所有点均可表示为具有“小”系数（以适当范数度量）的线性组合。形式上，对于向量集合 $X = \{v_1, v_2, \dots, v_n\}$，目标是找到 $T \subseteq [n]$ 使得每个 $v \in X$ 均可表示为 $\sum_{i\in T} \alpha_i v_i$，且 $\|\alpha\|$ 很小。这一概念（亦称良态基）已应用于多个领域，包括赌徒线性优化、行列式最大化及矩阵低秩近似。本文针对所有 $\ell_p$ 范数给出了体积极张量大小的近似最优界，并证明其可通过简单局部搜索算法构造。进而展示了该结果在其他任务中的应用，特别是最小体积包含椭球（MVEE）问题的核心集构造。