Semidefinite Approximations of the Matrix Logarithm
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1705.00812.pdf
Description
Accepted version
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575.21 KB
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Author(s)
Fawzi, Hamza
Saunderson, James
Parrilo, Pablo A.
Date Issued
March 21, 2018
Journal
Foundations of Computational Mathematics Foundations of Computational Mathematics
Publisher
Springer Nature
Citation
Fawzi, Hamza, et al. “Semidefinite Approximations of the Matrix Logarithm.” Foundations of Computational Mathematics 19, no. 2 (April 2019): 259–96.
Version
Author's final manuscript
Abstract
The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations. Keywords: Convex optimization, Matrix concavity, Quantum relative entropy
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s10208-018-9385-0
