标题： 齐次空间上的参数估计 显示英文标题

标题： Parameter Estimation on Homogeneous Spaces

摘要： 费雪信息度量（FIM）及其相关的克拉默-拉奥下界（CRB）是统计信号处理中的基本工具，用于指导实验和参数估计算法的有效设计。本文研究了当参数位于齐次空间上的情况时这些概念。与一般黎曼流形的现有费雪-拉奥理论不同，我们的重点是利用齐次空间的群论结构，这通常比它们的黎曼结构更容易处理。通过将齐次空间识别为陪集空间，我们描述了FIM的特性，提出了群论下的CRB及其推论，并澄清了它与黎曼CRB之间的关系。我们用工程中的两个例子来说明该理论的应用：（i）机器人姿态的估计和（ii）传感器网络定位。特别是，这些例子表明齐次空间提供了一个自然框架，用于研究对称群不变的统计模型。 显示英文摘要

摘要： The Fisher Information Metric (FIM) and the associated Cram\'er-Rao Bound (CRB) are fundamental tools in statistical signal processing, which inform the efficient design of experiments and algorithms for estimating the underlying parameters. In this article, we investigate these concepts for the case where the parameters lie on a homogeneous space. Unlike the existing Fisher-Rao theory for general Riemannian manifolds, our focus is to leverage the group-theoretic structure of homogeneous spaces, which is often much easier to work with than their Riemannian structure. The FIM is characterized by identifying the homogeneous space with a coset space, the group-theoretic CRB and its corollaries are presented, and its relationship to the Riemannian CRB is clarified. The application of our theory is illustrated using two examples from engineering: (i) estimation of the pose of a robot and (ii) sensor network localization. In particular, these examples demonstrate that homogeneous spaces provide a natural framework for studying statistical models that are invariant with respect to a group of symmetries.