研究扩展高斯族的希尔伯特几何，给出距离公式与不变性特性。
Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family
- 基于对称正定双锥体构造希尔伯特度量，推导闭式距离公式。
- 揭示该几何在参数变换下的不变性，适用于退化协方差情形。
- 为处理退化高斯分布提供新几何工具，适合统计建模研究者。
扩展高斯族是通过将退化协方差或退化精度矩阵（或两者混合）引入高斯族的闭包得到的。其参数空间构成一个对称半正定矩阵双锥体，即两个在基底处连接的半正定矩阵锥。本文研究该开有界凸对称正定双锥体的希尔伯特几何，给出对应希尔伯特度量距离的闭式表达式，并系统分析其不变性性质。还探讨了该几何在处理扩展高斯分布方面的潜在应用。
原文摘要 · Abstract (English)
The extended Gaussian family is the closure of the Gaussian family obtained by completing the Gaussian family with the counterpart elements induced by degenerate covariance or degenerate precision matrices, or a mix of both degeneracies. The parameter space of the extended Gaussian family forms a symmetric positive semi-definite matrix bicone, i.e. two partial symmetric positive semi-definite matrix cones joined at their bases. In this paper, we study the Hilbert geometry of such an open bounded convex symmetric positive-definite bicone. We report the closed-form formula for the corresponding Hilbert metric distance and study exhaustively its invariance properties. We also touch upon potential applications of this geometry for dealing with extended Gaussian distributions.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。