SIAM Journal on Computing, Volume 49, Issue 6, Page 1249-1270, January 2020.
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors $v_1,\ldots,v_m \in \mathbb{R}^d$ and a constraint family $\mathcal{B} \subseteq 2^{[m]}$, find a set $S \in \mathcal{B}$ that maximizes the squared volume of the simplex spanned by the vectors in $S$. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity, and partition or matroid constraints over $[m]$ are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint ($\mathcal{B}=(\begin{smallmatrix}{[m]} \\ {r}\end{smallmatrix})$, the problem becomes NP-hard and has received much attention starting with a result by Khachiyan [J. Complexity, 11 (1995) pp. 138--153] who gave an $r^{O(r)}$ approximation algorithm for a special case of this problem. Recently, Nikolov and Singh [Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, 2016, pp. 192--201] presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when $\mathcal{B}$ corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits [Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, ACM, 2006, pp. 417--426], and was recently extended to regular matroids [Straszak and Vishnoi, Proceedings of the Forty-ninth ACM SIGACT Symposium on Theory of Computing, 2017, pp. 370--383] and [Anaris and Gharan, Proceedings of the Forty-ninth Annual SIGACT Symposium on Theory of Computing, 2017, pp. 384--396] and general matroids [Anari, Gharan, and Vinzant, Proceedings of the Fifty-ninth IEEE Annual Symposium on Foundations of Computer Science, 2018, pp. 35--46]. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms---that also output a set---remained open. The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem, for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions, which allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena have recently been deployed.

中文翻译：

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通过非凸松弛和反集中实现子行列式最大化

SIAM计算学报，第49卷，第6期，第1249-1270页，2020年1月。
在优化和计算机科学中出现的几个基本问​​题可以解释为：给定向量$ v_1，\ ldots，v_m \ in \ mathbb {R} ^ d $和约束族$ \ mathcal {B} \ subseteq 2 ^ { [m]} $，在\ mathcal {B} $中找到一个集合$ S \，该集合使$ S $中的向量所覆盖的单纯形的平方体积最大。一个有启发性的例子是机器学习和信息检索中普遍存在的数据汇总问题，其中给出了一组代表诸如文档或图像之类的数据的特征向量。向量集合的体积被用作其多样性的度量，并且为了确保资源或公平性约束，对$ [m] $施加了分区或拟阵约束。即使有简单的基数约束（$ \ mathcal {B} =（\ begin {smallmatrix} {[m]} \\ {r} \ end {smallmatrix}）$，从卡奇扬的结果开始，这个问题就变成了NP难题，并引起了很多关注[J. 复杂性，第11卷，1995年，第138--153页]，他针对此问题的特殊情况给出了$ r ^ {O（r）} $近似算法。最近，Nikolov和Singh [2016年第四十八届ACM计算理论年会论文集，第192--201页]提出了一个凸程序，并说明了如何将其用于估算最多样化集的值。有多个基数约束（即，当$ \ mathcal {B} $对应于分区矩阵时）。他们对凸程序的完整性缺口的证明依赖于Gurvits的不等式[第38届ACM年度计算理论研讨会论文集，ACM，2006年，第417--426页]，最近扩展到了常规拟阵[Straszak和Vishnoi，第49届ACM SIGACT计算理论研讨会论文集，2017年，第370--383页]和[Anaris和Gharan，第49届SIGACT计算理论年度研讨会论文集，2017年，第384页- 396]和一般类人机器人[Anari，Gharan和Vinzant，《 IEEE第59届计算机科学基础年会论文集》，2018年，第35--46页]。这些估计算法是否可以转换为更有用的近似算法（也输出​​一个集合）的问题仍然存在。本文的主要贡献是为分区和常规拟阵给出了第一个近似算法。对于这些拟阵，我们提出了行列式最大化问题的新公式；这将它们简化为以下问题：找到一个在概率单纯形的笛卡尔积上最大化非凸函数的绝对值的点。我们的结果的技术核心是针对这些函数引起的因变量的新的反浓度不等式，这使我们能够将这些非凸函数的最优值与其在随机点的值相关联。与关于受约束的行列式最大化问题的先前工作不同，我们的证明不依赖于实际稳定性或凸性，并且在最近已部署反集中现象的算法和复杂性上都可能具有独立的兴趣。这使我们能够将这些非凸函数的最优值与它们在随机点处的值相关联。与关于受约束的行列式最大化问题的先前工作不同，我们的证明不依赖于实际稳定性或凸性，并且在最近已部署反集中现象的算法和复杂性上都可能具有独立的兴趣。这使我们能够将这些非凸函数的最优值与它们在随机点处的值相关联。与关于受约束的行列式最大化问题的先前工作不同，我们的证明不依赖于实际稳定性或凸性，并且在最近已部署反集中现象的算法和复杂性上都可能具有独立的兴趣。