Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bures-Wasserstein minimizing geodesics between covariance matrices of
  different ranks   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 $\Sigma$ and $\Lambda$ 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 $\Sigma$, $\Lambda$, $\Sigma$$\Lambda$. In
particular, the minimizing geodesic is unique if and only if $r = \min(k, l)$.
Otherwise, there are infinitely many.

3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph