正定算子的Wasserstein重心
On the Wasserstein barycenter of positive definite operators
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中文总结 AI 辅助
该研究将正定矩阵的Bures-Wasserstein均值推广到希尔伯特空间正定算子情形，证明其存在唯一性，分析其生成的ODE半群的指数收缩性，还证明相关极限定理并建立算子不等式。
中文摘要 AI 辅助
我们将正定矩阵的Bures-Wasserstein均值推广到希尔伯特空间上正定算子的情形，这通过其定义的驻点算子方程实现，该方程来自由正定矩阵表示的中心化高斯分布的平方Bures-Wasserstein距离之和的梯度。我们证明该梯度具有Fréchet导数，其在希尔伯特-施密特算子空间上诱导出一个有界线性算子，该算子具有严格正实谱。通过将其延拓到一般有界线性算子时的谱持久性，我们可得出该均值的存在性与唯一性，同时这也使得研究其生成的ODE半群成为可能，这类半群在通过等价重赋范构造得到的Banach-Finsler度量下具有指数收缩性。利用该流的指数收缩性，我们证明了一类“Nodice”型定理及其随机变体，即支撑有界的概率测度的Sturm型强大数定律。我们还验证了该均值的基本性质，并建立了其满足的各类算子不等式。
英文摘要
We extend the Bures-Wasserstein mean of positive definite matrices to the case of positive definite operators on a Hilbert space. This is done through its defining stationary point operator equation, coming from the gradient of the sum of squared Bures-Wasserstein distances of centered Gaussians represented by positive definite matrices. This gradient is shown to have a Fréchet derivative which induces a bounded linear operator on the space of Hilbert-Schmidt operators with strictly positive real spectrum. This allows us to conclude the existence and uniqueness of this mean by exhibiting the spectral permanence of this operator when extended to general bounded linear operators and also enables the study of its generated ODE semigroups, which enjoy exponential contraction in a Banach-Finsler metric obtained through the construction of equivalent renormings. Using this exponential contractivity of the flow, we prove a `Nodice'-type of theorem and its stochastic variant, a Sturm-type of strong law of large numbers for probability measures with bounded support. We also verify fundamental properties and establish various operator inequalities satisfied by the Wasserstein mean.
