Positive definite matrices and the S-divergence
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Abstract:
Hermitian positive definite (hpd) matrices form a self-dual convex cone whose interior is a Riemannian manifold of nonpositive curvature. The manifold view comes with a natural distance function but the conic view does not. Thus, drawing motivation from convex optimization we introduce the S-divergence, a distance-like function on the cone of hpd matrices. We study basic properties of the S-divergence and explore its connections to the Riemannian distance. In particular, we show that (i) its square-root is a distance, and (ii) it exhibits numerous nonpositive-curvature-like properties.References
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Additional Information
- Suvrit Sra
- Affiliation: Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 815361
- Email: suvrit@mit.edu
- Received by editor(s): March 6, 2015
- Received by editor(s) in revised form: August 18, 2015
- Published electronically: October 22, 2015
- Additional Notes: This work was done while the author was with the MPI for Intelligent Systems, Tübingen, Germany. A small fraction of this work was presented at the Neural Information Processing Systems (NIPS) Conference 2012.
- Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2787-2797
- MSC (2010): Primary 15A45, 52A99, 47B65, 65F60
- DOI: https://doi.org/10.1090/proc/12953
- MathSciNet review: 3487214