The Derivative of the Matrix Geometric Mean with an Application to the Nonnegative Decomposition of Tensor Grids

We provide an expression for the derivative of the weighted matrix geometric mean, with respect to both the matrix arguments and the weights, that can be easily translated to an algorithm for its computation. As an application, we consider the problem of the approximate decomposition of a tensor grid M, a matrix whose entries are positive definite matrices. For different geometries on the set of positive definite matrices, we derive an approximate decomposition such that any column of M is a barycentric combination of the columns of a smaller tensor grid. This extends the Euclidean case, already considered in the literature, to the geometry in which the barycenter is the matrix geometric mean and the log-Euclidean geometry.