发表机构
Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences; Moscow Institute of Physics and Technology; Skolkovo Institute of Science and Technology（俄罗斯科学院马丘克数值数学研究所; 莫斯科物理技术学院; 斯科尔科沃科学技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对矩阵子集选择问题，提出前向去随机化体积采样算法（FDVS），以 $O(nkm)$ 复杂度达到已知最优期望界，并通过实验验证其在最优实验设计等任务中的有效性。
AI 中文摘要
我们研究矩阵的子集选择问题，目标是从一个“短胖”矩阵 $X \in \mathbb{R}^{m \times n}$ 中选取 $k$ 个列索引的子集 $\mathcal{S}$，使得采样后的子矩阵 $X_{\mathcal{S}}$ 满足 $\\|X_{\mathcal{S}}^†X\\|_F$ 尽可能小。我们的方法以体积采样为核心，该方法在期望意义上达到了该目标已知的最紧界。作为主要贡献，我们提出了一种新的确定性算法——前向去随机化体积采样（FDVS），该算法可证明达到此界，且渐近复杂度为 $O(nkm)$。相比之下，当 $k \ll n$ 时，所有先前已知的具有此保证的算法复杂度在 $n$ 上至少为二次增长，这使得 FDVS 对极宽矩阵特别有吸引力。此外，我们系统地梳理了体积采样方法的全景：我们提出了一种简单的 $O(nm^2)$ 精确前向体积采样算法，证明了一种已知的确定性方法实际上是反向迭代体积采样的去随机化，并推导了后者的一个改进版本以避免重复的 SVD 降阶更新，同时提出了一个快速的贪心前向变体。我们在涉及最优实验设计、传感器布置和 DLRA-DEIM 的数值实验中验证并比较了这些算法。
英文摘要
We address the Subset selection problem for matrices, where the goal is to select a subset $\mathcal{S}$ of $k$ column indices from a \enquote{short-and-fat} matrix $X \in \mathbb{R}^{m \times n}$, such that the sampled submatrix $X_{\mathcal{S}}$ has $\|X_{\mathcal{S}}^†X\|_F$ as small as possible. Our approach is centered on volume sampling, which attains the tightest known bound on this objective in expectation. As our primary contribution, we propose a new deterministic algorithm, Forward derandomized volume sampling (FDVS), which provably attains this bound and has asymptotic complexity $O(nkm)$. In contrast, the complexity of all previously known algorithms with this guarantee scales at least quadratically in $n$ when $k \ll n$, making FDVS particularly attractive for very wide matrices. In addition, we systematically structure the landscape of volume sampling methods: we propose a simple $O(nm^2)$ algorithm for exact forward volume sampling, show that a known deterministic method is in fact a derandomization of Reverse iterative volume sampling, and derive a modification of the latter that avoids repeated SVD downdating, along with a fast greedy forward variant. The algorithms are verified and compared in numerical experiments involving optimal experimental design, sensor placement, and DLRA-DEIM.
Comments47 pages, 7 figures
