Weighted inductive means

Abstract

In this paper we present a unified framework for weighted inductive means on the cone P of positive definite Hermitian matrices as natural multivariable extensions of two variable weighted means, particularly of metric midpoint operations on P. It includes some well-known multivariable weighted matrix means: the weighted arithmetic, harmonic, resolvent, Sturm's inductive geometric mean on the Riemannian manifold P equipped with the trace metric, Log-Euclidean and spectral geometric means. A recursion (or weight additive) formula is derived and applied to find a closed form and basic properties for a weighted inductive mean. An upper bound on the sensitivity, a metric characterization and min and max optimization problems over permutations for the inductive geometric mean are presented. Moreover, we apply the obtained results to a class of midpoint operations of the non-positively curved Hadamard metrics on P parameterized over Hermitian unitary matrices.