Semidefinite approximations of matrix logarithm

The matrix logarithm, when applied to symmetric positive definite matrices satisfies a notable concavity property in the positive semidefinite (Loewner) order. This concavity property is a cornerstone result in the study of operator convex functions and has important applications in matrix concentration inequalities and quantum information theory.
In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. These approximations are also useful in the scalar case and provide a much faster alternative to existing methods based on successive approximation for problems involving the exponential/relative entropy cone. I will conclude by showing some applications to problems arising in quantum information theory.

This is joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT)

• Speaker: Hamza Fawzi, University of Cambridge
• Thursday 03 November 2016, 14:00–15:00
• Venue: Cambridge University Engineering Department, JDB Seminar Room.
• Series: CUED Control Group Seminars; organiser: Tim Hughes.