关于具有给定重心且被标准单纯形包围的单纯形
On Simplices with a Given Barycenter That Are Enclosed by the Standard Simplex
AI总结：
本文提出在随机矩阵流形上用迹最大化替代行列式最大化来求解最大体积单纯形问题，证明了解的最优性、唯一性条件及其逆M-矩阵性质，并应用于合成条件累积分布函数。
摘要：
我们提出了一个定义在随机矩阵集合流形上的优化模型。从几何角度看，该模型类似于识别一个具有给定重心且被 n 维标准单纯形包围的最大体积 n 维单纯形。最大化单纯形的体积等价于最大化其对应矩阵的行列式。在我们的模型中，我们采用迹最大化作为行列式最大化的线性替代方案。我们确定了该模型一个解的解析形式。我们证明该解是最优的，并给出了它是唯一最优解的充分必要条件。此外，我们证明了所确定的最优解是一个逆 M-矩阵，并且其特征值与其对角线元素相同。我们展示了该模型及其解如何应用于合成条件累积分布函数（CDF）的任务，这些条件 CDF 与给定的离散边际分布协同作用，能够一致地保留给定的 CDF。
英文摘要：
We present an optimization model defined on the manifold of the set of stochastic matrices. Geometrically, the model is akin to identifying a maximum-volume $n$-dimensional simplex that has a given barycenter and is enclosed by the $n$-dimensional standard simplex. Maximizing the volume of a simplex is equivalent to maximizing the determinant of its corresponding matrix. In our model, we employ trace maximization as a linear alternative to determinant maximization. We identify the analytical form of a solution to this model. We prove the solution is optimal and present necessary and sufficient conditions for it to be the unique optimal solution. Additionally, we show the identified optimal solution is an inverse $M$-matrix, and that its eigenvalues are the same as its diagonal entries. We demonstrate how the model and its solutions apply to the task of synthesizing conditional cumulative distribution functions (CDFs) that, in tandem with a given discrete marginal distribution, coherently preserve a given CDF.
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。
