论文标题
Bures-Wasserstein最小化不同等级的协方差矩阵之间的测量学
Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks
论文作者
论文摘要
配备Bures-Wasserstein距离的协方差矩阵是正交组在方形矩阵的欧几里得空间上平滑，正确和等距作用的轨道空间。这种结构在协方差矩阵上引起了天然轨道分层，这正是等级的分层。因此，地层是固定等级的对称阳性半明确（PSD）矩阵的歧管，该矩阵赋予了Bures-Wasserstein Riemannian度量。在这项工作中，我们研究了Bures-Wasserstein距离的测量学。首先，我们通过阐明指数图的预映率并指定注入域，从而完成了每个层中大地测量学的文献。我们还提供了水平升降机，指数图和Riemannian对数的明确公式，这些配方在以前的作品中被隐含。其次，我们给出所有最小化的测量段的表达，这些片段与任何等级的两个协方差矩阵。更确切地说，我们表明，所有最小化的测量学矩阵$σ$和$λ$之间的所有最小化大地测量的集合是由$ \ Mathbb {r}^{（k-r）\ times（l-r）\ times（l-r）$的封闭单位球对频谱规范的$ k，l，r $ n $ k，l，r $是$ $ q $ c $ c $ c，$ k，r）$ $ c，$ k，l r）$ $。特别是，当且仅当$ r = \ min（k，l）$时，最小化的测量是唯一的。否则，会有很多。
The set of covariance matrices equipped with the Bures-Wasserstein distance is the orbit space of the smooth, proper and isometric action of the orthogonal group on the Euclidean space of square matrices. This construction induces a natural orbit stratification on covariance matrices, which is exactly the stratification by the rank. Thus, the strata are the manifolds of symmetric positive semi-definite (PSD) matrices of fixed rank endowed with the Bures-Wasserstein Riemannian metric. In this work, we study the geodesics of the Bures-Wasserstein distance. Firstly, we complete the literature on geodesics in each stratum by clarifying the set of preimages of the exponential map and by specifying the injection domain. We also give explicit formulae of the horizontal lift, the exponential map and the Riemannian logarithms that were kept implicit in previous works. Secondly, we give the expression of all the minimizing geodesic segments joining two covariance matrices of any rank. More precisely, we show that the set of all minimizing geodesics between two covariance matrices $Σ$ and $Λ$ is parametrized by the closed unit ball of $\mathbb{R}^{(k-r)\times(l-r)}$ for the spectral norm, where $k, l, r$ are the respective ranks of $Σ$, $Λ$, $Σ$$Λ$. In particular, the minimizing geodesic is unique if and only if $r = \min(k, l)$. Otherwise, there are infinitely many.
