Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. (English) Zbl 1220.47025
This paper extends the natural metric for positive definite \(n\times n\) (full rank) matrices to the set \(S^+(p,n)\) of positive semidefinite matrices of rank \(p\leq n\). One of the main properties of the former is invariance under the action of the general linear group \(GL(n)\) by congruence. The metric defined here for \(S^+(p,n)\) is not (nor could be) invariant for all invertible transformations, but is preserved by rotations and scalings.
The manifold \(S^+(p,n)\) is proved to be geodesically complete, and special curves which approximate geodesics are computed.
A new definition of geometric mean between matrices of rank \(p\) is derived, which preserves the rank and, moreover, is computable.
The authors argue that this metric provides computational tools which might prove useful in the application, such as MRI tensor computing, radar processing, machine learning and bio-informatics.
The manifold \(S^+(p,n)\) is proved to be geodesically complete, and special curves which approximate geodesics are computed.
A new definition of geometric mean between matrices of rank \(p\) is derived, which preserves the rank and, moreover, is computable.
The authors argue that this metric provides computational tools which might prove useful in the application, such as MRI tensor computing, radar processing, machine learning and bio-informatics.
Reviewer: Esteban Andruchow (Los Polvorines)
MSC:
| 47A64 | Operator means involving linear operators, shorted linear operators, etc. |
| 53C22 | Geodesics in global differential geometry |
| 58D19 | Group actions and symmetry properties |
| 54B15 | Quotient spaces, decompositions in general topology |
| 53C20 | Global Riemannian geometry, including pinching |
