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Positive definite matrices and the S-divergence


Author: Suvrit Sra
Journal: Proc. Amer. Math. Soc. 144 (2016), 2787-2797
MSC (2010): Primary 15A45, 52A99, 47B65, 65F60
Published electronically: October 22, 2015
MathSciNet review: 3487214
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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.


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Additional Information

Suvrit Sra
Affiliation: Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: suvrit@mit.edu

DOI: https://doi.org/10.1090/proc/12953
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
Article copyright: © Copyright 2015 American Mathematical Society