最大熵采样问题的最优主子矩阵选择：可扩展算法与性能保证
Best Principal Submatrix Selection for the Maximum Entropy Sampling Problem: Scalable Algorithms and Performance Guarantees
AI总结：
本文针对最大熵采样问题，提出新型凸整数规划，推导改进的采样算法近似界，给出确定性实现与局部搜索算法的首个近似界及高效实现，实验验证算法可高效求解中大规模实例，还拓展至A-最优MESP并开源算法。
摘要：
本文研究经典的最大熵采样问题（MESP），该问题旨在从协方差矩阵中选择预设大小的最具信息量的主子矩阵。MESP已被广泛应用于医疗保健、电力系统、制造业和数据科学等诸多领域。通过研究其拉格朗日对偶与原问题表征，我们推导得出一种针对MESP的新型凸整数规划，并证明其连续松弛可得到近似最优解。这些结果促使我们研究一种高效的采样算法，并推导了其针对MESP的近似界，该界改进了现有文献中已知的最优界。随后，我们给出了该采样算法的一种高效确定性实现，其具备相同的近似界。通过为奇异矩阵开发新的数学工具，并分析所提出凸整数规划的拉格朗日对偶，我们研究了被广泛使用的局部搜索算法，并证明了其针对MESP的首个已知近似界。这些证明技术进一步启发了我们设计局部搜索算法的高效实现方式。我们的数值实验表明，这些近似算法能够高效地将中等规模和大规模实例求解至近似最优状态。我们提出的算法已被编码并作为开源软件发布。最后，我们将分析拓展至A-最优MESP（A-MESP），该问题的目标是最小化所选主子矩阵的逆的迹。
英文摘要：
This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program for MESP and show that its continuous relaxation yields a near-optimal solution. The results motivate us to study an efficient sampling algorithm and develop its approximation bound for MESP, which improves the best-known bound in literature. We then provide an efficient deterministic implementation of the sampling algorithm with the same approximation bound. By developing new mathematical tools for the singular matrices and analyzing the Lagrangian dual of the proposed convex integer program, we investigate the widely-used local search algorithm and prove its first-known approximation bound for MESP. The proof techniques further inspire us with an efficient implementation of the local search algorithm. Our numerical experiments demonstrate that these approximation algorithms can efficiently solve medium-sized and large-scale instances to near-optimality. Our proposed algorithms are coded and released as open-source software. Finally, we extend the analyses to the A-Optimal MESP (A-MESP), where the objective is to minimize the trace of the inverse of the selected principal submatrix.
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。
