关于矩阵值测度的最优输运

可分离度量空间上矩阵值核的 Mercer-Young 定理

The Mercer-Young Theorem for Matrix-Valued Kernels on Separable Metric Spaces

将 Mercer-Young 正定核刻画定理推广至可分离度量空间上的矩阵值核，并展示其在凸优化等领域的应用。

A mass transport approach to the optimization of adapted couplings of real valued random variables

In this work, we investigate an optimization problem over adapted couplings between pairs of real valued random variables, possibly describing random times. We relate those couplings to a specific class of causal transport plans between probabilities on the set of real numbers endowed with a filtration, for which their provide a specific representation, which is motivated by a precise characterization of the corresponding deterministic transport plans. From this, under mild hypothesis, the existence of a solution to the problem investigated here is obtained. Furthermore, several examples are p…

线性矩阵不等式约束下矩阵值函数秩和惯量的最优化问题的解析解

Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities

矩阵秩和惯性最优化问题是一类不连续的最优化问题，其中决策变量是在一定可行矩阵集上运行的矩阵，而变量矩阵的秩和惯性是整值目标函数。本文建立了Löwner偏序中埃尔米特矩阵表达式$A_1 - B_1XB_1^{*}$满足线性矩阵不等式$B_2XB_2^{*} \succcurlyeq A_2$ $(B_2XB_2^{*} \preccurlyeq A_2)$的秩目标函数和惯性目标函数的最大值和最小值的一组显式计算公式，并给出了这些公式在某些约束矩阵值函数的性质刻画中的应用.

On the stability of the martingale optimal transport problem: A set-valued map approach

Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [21]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck, Jourdain, Margheriti, and Pammer [5], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w…

矩阵值Riccati扩散的稳定性

On the stability of matrix-valued Riccati diffusions

研究了矩阵值Riccati扩散的稳定性。在这项工作中考虑的矩阵值Riccati扩散过程的兴趣在自己的权利，作为一个相当典型的模型的矩阵值二次随机过程。在相当自然的能观性和能控性条件下，我们得到了时间一致矩和波动估计以及指数收缩不等式。我们的方法结合了谱理论，非线性半群方法和随机阵微积分。这是这类矩阵值随机微分方程的首次分析。这类随机模型出现在信号处理和数据同化中，特别是在集合卡尔曼-布西滤波理论中。在这种情况下，Riccati扩散表示与McKean-Vlasov型交互卡尔曼-布西滤波器相关联的样本协方差阵的流动。这里开发的分析适用于不稳定信号的滤波问题。

关于$F_{LG}$矩阵的单位行和与实值性质

On the Unity Row Summation and Real Valued Nature of the $F_{LG}$ Matrix

电力系统的计算在很大程度上依赖于$Y_{bus}$矩阵，它是所研究的网络的拉普拉斯矩阵，由每个分支的复值导纳加权。将$Y_{bus}$划分为四个子矩阵以分别量化网络中负载和发电节点之间的连通性通常是有用的。这些子矩阵的简单操作给出了$F_{LG}$矩阵，该矩阵提供了关于电压偏差如何通过电力系统传播以及如何最小化能量损失的有用见解。不同的作者已经观察到，在实践中的元素$F_{LG}$是实值和它的行总和接近1：本文件解释和证明这些属性。

向量值测度的非平衡 L1 最优传输及其在全波形反演中的应用

Unbalanced L1 optimal transport for vector valued measures and application to Full Waveform Inversion

本文提出了一种基于 $L^1$ 的向量值测度非平衡最优传输方法，通过将数据提升至高维自对偶锥恢复有意义的传输问题，继承了 $L^1$ 的计算优势，并验证了其在全波形地震反演中的潜力。

Optimal transport for vector Gaussian mixture models

Vector-valued Gaussian mixtures form an important special subset of vector-valued distributions. In general, vector-valued distributions constitute natural representations for physical entities, which can mutate or transit among alternative manifestations distributed in a given space. A key example is color imagery. In this note, we vectorize the Gaussian mixture model and study several different optimal mass transport related problems associated to such models. The benefits of using vector Gaussian mixture for optimal mass transport include computational efficiency and the ability to preserve…

矩阵值观测的低秩凸聚类

Low Rank Convex Clustering For Matrix-Valued Observations

针对矩阵值观测的低秩特性，提出低秩凸聚类模型，扩展了向量数据方法并作为低秩k-means的凸松弛，理论证明精确簇恢复与预测一致性，并开发高效双循环算法验证其有效性。

计算矩阵值函数 $A + BXC$ 在 $X$ 秩固定时的极值秩的解析公式

Analytical formulas for calculating the extremal ranks of the matrix-valued function $A + BXC$ when the rank of $X$ is fixed

本文利用 $A$, $B$, $C$ 的同步分解和初步结果，建立了当变量矩阵 $X$ 的秩固定时，$A + BXC$ 的最大和最小秩的解析公式，并应用于部分指定分块矩阵的秩补全。