Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family

ML BAY PGM
489
2025年08月20日
扩展高斯族是通过对高斯族进行完备化而得到的，这种完备化是通过加入由退化的协方差矩阵或退化的精度矩阵（或两者的混合退化情况）所诱导出的对应元素来实现的。扩展高斯族的参数空间构成了一个对称的半正定矩阵双锥体，即两个部分对称的半正定矩阵锥在底面连接在一起的结构。本文研究了这样一个开的、有界的、凸的对称正定双锥体的希尔伯特几何结构。我们给出了对应的希尔伯特度量距离的闭合表达式，并详尽地研究了它的不变性性质。此外，我们还探讨了该几何结构在处理扩展高斯分布时的潜在应用。
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.
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