The Kähler Mean of Block-Toeplitz Matrices with Toeplitz Structured Blocks

When computing an average of positive definite (PD) matrices, the preservation of additional matrix structure is desirable for interpretations in applications. An interesting and widely present structure is that of PD Toeplitz matrices, which we endow with a geometry originating in signal processing theory. As an averaging operation, we consider the barycenter, or minimizer of the sum of squared intrinsic distances. The resulting barycenter, the Kähler mean, is discussed along with its origin. Also, a generalization of the mean towards PD (Toeplitz-Block) Block-Toeplitz matrices is discussed. For PD Toeplitz-Block Block-Toeplitz matrices, we derive the generalized barycenter, or generalized Kähler mean, and a greedy approximation. This approximation is shown to be close to the generalized mean with a significantly lower computational cost.