# Semidefinite Approximations of the Matrix Logarithm

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## 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## Mathematics Subject Classification

90C22 52A41 47A63## References

- 1.Al-Mohy, A.H., Higham, N.J.: Improved inverse scaling and squaring algorithms for the matrix logarithm. SIAM J. Sci. Comput.
**34**(4), C153–C169 (2012)MathSciNetCrossRefMATHGoogle Scholar - 2.ApS, M.: The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28). (2015). http://docs.mosek.com/7.1/toolbox/index.html
- 3.Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications. SIAM (2001)Google Scholar
- 4.Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res.
**26**(2), 193–205 (2001)MathSciNetCrossRefMATHGoogle Scholar - 5.Besenyei, A., Petz, D.: Successive iterations and logarithmic means. Oper. and Matrices
**7**(1), 205–218 (2013). https://doi.org/10.7153/oam-07-12 - 6.Bhatia, R.: Positive definite matrices. Princeton University Press (2009)Google Scholar
- 7.Bhatia, R.: Matrix analysis, vol. 169. Springer Science & Business Media (2013)Google Scholar
- 8.Blekherman, G., Parrilo, P.A., Thomas, R.R.: Semidefinite optimization and convex algebraic geometry. SIAM (2013)Google Scholar
- 9.Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng.
**8**(1), 67–127 (2007)MathSciNetCrossRefMATHGoogle Scholar - 10.Bushell, P.J.: Hilbert’s metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal.
**52**(4), 330–338 (1973)MathSciNetCrossRefMATHGoogle Scholar - 11.Carlen, E.A.: Trace inequalities and quantum entropy. An introductory course. In: Entropy and the quantum, vol. 529, pp. 73–140. AMS (2010)Google Scholar
- 12.Carlson, B.C.: An algorithm for computing logarithms and arctangents. Math. Comp.
**26**(118), 543–549 (1972)MathSciNetCrossRefMATHGoogle Scholar - 13.Cox, D.A.: The arithmetic-geometric mean of Gauss. In: Pi: A source book, pp. 481–536. Springer (2004)Google Scholar
- 14.Dieci, L., Morini, B., Papini, A.: Computational techniques for real logarithms of matrices. SIAM J. Matrix Anal. Appl.
**17**(3), 570–593 (1996)MathSciNetCrossRefMATHGoogle Scholar - 15.Domahidi, A., Chu, E., Boyd, S.: ECOS: An SOCP solver for embedded systems. In: European Control Conference (ECC), pp. 3071–3076 (2013)Google Scholar
- 16.Ebadian, A., Nikoufar, I., Gordji, M.E.: Perspectives of matrix convex functions. Proc. Natl. Acad. Sci. USA
**108**(18), 7313–7314 (2011)MathSciNetCrossRefMATHGoogle Scholar - 17.Effros, E., Hansen, F.: Non-commutative perspectives. Ann. Funct. Anal
**5**(2), 74–79 (2014)MathSciNetCrossRefMATHGoogle Scholar - 18.Effros, E.G.: A matrix convexity approach to some celebrated quantum inequalities. Proc. Natl. Acad. Sci. USA
**106**(4), 1006–1008 (2009)MathSciNetCrossRefMATHGoogle Scholar - 19.Fawzi, H., Fawzi, O.: Relative entropy optimization in quantum information theory via semidefinite programming approximations. arXiv preprint arXiv:1705.06671 (2017)
- 20.Fawzi, H., Saunderson, J.: Lieb’s concavity theorem, matrix geometric means, and semidefinite optimization. Linear Algebra Appl.
**513**, 240–263 (2017)MathSciNetCrossRefMATHGoogle Scholar - 21.Fujii, J., Kamei, E.: Relative operator entropy in noncommutative information theory. Math. Japon
**34**, 341–348 (1989)MathSciNetMATHGoogle Scholar - 22.Glineur, F.: Quadratic approximation of some convex optimization problems using the arithmetic-geometric mean iteration. Talk at the “Workshop GeoLMI on the geometry and algebra of linear matrix inequalities”. http://homepages.laas.fr/henrion/geolmi/geolmi-glineur.pdf (retrieved November 2, 2016) (2009)
- 23.Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2014)
- 24.Hansen, F., Pedersen, G.K.: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann.
**258**(3), 229–241 (1982)MathSciNetCrossRefMATHGoogle Scholar - 25.Helton, J.W., Klep, I., McCullough, S.: The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. J. Eur. Math. Soc. (JEMS)
**19**(6), 1845–1897 (2017)MathSciNetCrossRefMATHGoogle Scholar - 26.Higham, N.J.: Functions of matrices: theory and computation. SIAM (2008)Google Scholar
- 27.Kenney, C., Laub, A.J.: Condition estimates for matrix functions. SIAM J. Matrix Anal. Appl.
**10**(2), 191–209 (1989)MathSciNetCrossRefMATHGoogle Scholar - 28.Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math.
**11**(3), 267–288 (1973)MathSciNetCrossRefMATHGoogle Scholar - 29.Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum mechanical entropy. J. Math. Phys.
**14**(12), 1938–1941 (1973)MathSciNetCrossRefGoogle Scholar - 30.Meurant, G., Sommariva, A.: Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab. Numerical Algorithms
**67**(3), 491–506 (2014)MathSciNetCrossRefMATHGoogle Scholar - 31.Nesterov, Y.E.: Constructing self-concordant barriers for convex cones. CORE Discussion Paper (2006/30) (2006)Google Scholar
- 32.Sagnol, G.: On the semidefinite representation of real functions applied to symmetric matrices. Linear Algebra Appl.
**439**(10), 2829–2843 (2013)MathSciNetCrossRefMATHGoogle Scholar - 33.Serrano, S.A.: Algorithms for unsymmetric cone optimization and an implementation for problems with the exponential cone. Ph.D. thesis, Stanford University (2015)Google Scholar
- 34.Skajaa, A., Ye, Y.: A homogeneous interior-point algorithm for nonsymmetric convex conic optimization. Math. Program.
**150**(2), 391–422 (2015)MathSciNetCrossRefMATHGoogle Scholar - 35.Stoer, J., Bulirsch, R.: Introduction to numerical analysis, vol. 12, 3 edn. Springer-Verlag, New York (2002)CrossRefMATHGoogle Scholar
- 36.Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev.
**50**(1), 67–87 (2008)MathSciNetCrossRefMATHGoogle Scholar - 37.Trefethen, L.N.: Approximation theory and approximation practice. SIAM (2013)Google Scholar
- 38.Tropp, J.A.: From joint convexity of quantum relative entropy to a concavity theorem of Lieb. Proc. Amer. Math. Soc.
**140**(5), 1757–1760 (2012)MathSciNetCrossRefMATHGoogle Scholar - 39.Tropp, J.A.: An introduction to matrix concentration inequalities. Found. Trends Mach. Learn.
**8**(1-2), 1–230 (2015)CrossRefMATHGoogle Scholar