发表机构
Federal University of Rio de Janeiro; University of Michigan（里约热内卢联邦大学; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对三对角协方差矩阵的最大熵采样问题，提出更快的动态规划算法，将时间复杂度从O(n^5)降至O(ns^2)或O(ns)，并给出闭式解，同时扩展至蜘蛛图结构。
AI 中文摘要
最大熵采样问题（MESP）旨在为阶数为$n$的协方差矩阵$C$寻找一个阶数为$s$且具有最大对数行列式的主子矩阵。为了方便起见，我们假设$C$是非奇异的。Al-Thani和Lee（2023）在$C$或$C^{-1}$为三对角矩阵时，以$O(n^5)$的时间复杂度解决了MESP。我们证明他们的递归中的内部最大化仅依赖于索引集的前缀，并且解的任何一部分的长度不超过$s$；这给出了一个$O(ns^2)$时间的算法，该算法为每个预算$t\le s$返回最优值。当$C^{-1}$为三对角矩阵时，$C$在缩放意义下是Ornstein--Uhlenbeck过程在不等间隔时间点观测到的协方差矩阵，此时MESP变为在一条直线上选择点，其凹间隙函数具有Monge性质；这给出了一个$O(ns)$时间的算法，并且在一阶自回归情形下，有闭式解。仅给定$C$，当$C$或$C^{-1}$为三对角矩阵（允许对称置换）时，我们在$O(n^2)$时间内求解MESP，并在相同界限内识别这些情形。对于蜘蛛图，我们明确并锐化了运行时间对腿数的依赖，并借鉴Ohsaka关于星图的困难性结果，我们观察到，除非$\mathrm{P}=\mathrm{NP}$，否则运行时间的指数必须随腿数增长。
英文摘要
The maximum-entropy sampling problem (MESP) seeks, for an order-$n$ covariance matrix $C$, a principal submatrix of order $s$ with maximum log-determinant. Mostly for convenience, we assume that $C$ is nonsingular. Al-Thani and Lee (2023) solved MESP in $O(n^5)$ time when $C$ or $C^{-1}$ is tridiagonal. We show that the inner maximization of their recursion depends only on a prefix of the index set and that no piece of a solution is longer than $s$; this gives an $O(ns^2)$-time algorithm that returns the optimal value for every budget $t\le s$. When $C^{-1}$ is tridiagonal, $C$ is, up to scaling, the covariance matrix of an Ornstein--Uhlenbeck process observed at unevenly spaced times, and MESP becomes choosing points on a line under a concave gap function with the Monge property; this gives an $O(ns)$-time algorithm and, in the first-order autoregressive case, a closed-form solution. Given only $C$, we solve MESP in $O(n^2)$ time whenever $C$ or $C^{-1}$ is tridiagonal, up to a symmetric permutation, and we recognize these cases within the same bound. For spiders, we make explicit, and sharpen, the dependence on the number of legs, and, drawing on a hardness result of Ohsaka for stars, we observe that, unless $\mathrm{P}=\mathrm{NP}$, the exponent of the running time must grow with the number of legs.
