Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank

By Bart Vandereycken, Pierre-Antoine Absil, Stefan Vandewalle. 2009 IEEE/SP 15th Workshop on Statistical Signal Processing, 2009

62 citations recorded by the source corpus.

This paper deals with the Riemannian geometry of the set of symmetric positive semidefinite matrices of fixed rank. This set is studied as an embedded submanifold of the real matrices equipped with the usual Euclidean metric. With this structure, we derive expressions of the tangent space and geodesics of the manifold, suitable for efficient numerical computations.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1d7733bad78d482789d7b44e63a3365c2aea551e

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In Applied Mathematics

In Astronomy and Astrophysics

In Computational Theory and Mathematics

Citations in the corpus

Stabilization of projection‐based reduced‐order models (2012, background): “R(k+p)×(k+p) of dimension d1 = k(k + p)− k(k−1) 2 [21], and the set SPD(k + p) is an embedded manifold of R(k+p)×(k+p) of dimension d2 = (k+p)(k+p+1) 2 .”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, methodology): “…geometries for S+(p, n): a submanifold embedded in Rn×n as in Helmke & Moore (1994), Helmke & Shayman (1995), Orsi et al. (2004), Orsi et al. (2006), Vandereycken et al. (2009), Vandereycken & Vandewalle (2010); a quotient manifold of Rn×p as in Absil et al. (2009), Journée et al. (2010), Meyer…”