Means of Hermitian positive-definite matrices based on the log-determinant α-divergence function

The set of Hermitian positive-definite matrices plays fundamental roles in many disciplines such as mathematics, numerical analysis, probability and statistics, engineering, and biological and social sciences. In the last few years, there has been a renewable interest in developing the theory of means for elements in this set. This is due to theoretical and practical implications. In this work we present a one-parameter family of divergence functions for measuring distances between Hermitian positive-definite matrices. We then study the invariance properties of these divergence functions as well as the matrix means based on them. We also give globally convergent algorithms for computing these means.