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Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks.(English)Zbl 1528.15027

The authors consider the set of all positive semidefinite \(n\times n\) matrices (known to statisticians ascovariance matrices), for any integer \(n\). They equip this set with the Bures-Wasserstein metric, a particular Riemannian metric which is invariant under orthogonal change of basis. They compute the domain of the exponential map, the logarithms, horizontal lifts, and the minimizing geodesics between any two points. They overcome significant and inevitable technicalities, since the space of covariance matrices of given size \(n\) is not a manifold, but is naturally stratified by rank.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A63 Quadratic and bilinear forms, inner products
53B20 Local Riemannian geometry
53C22 Geodesics in global differential geometry
58D17 Manifolds of metrics (especially Riemannian)
54E50 Complete metric spaces
58A35 Stratified sets

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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