Dual-connection midpoint matrix means and their Gauss composition

Abstract

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic--harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We consider the more general problem of when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection determine three midpoint maps associated with the connection, its metric dual, and the Levi--Civita connection. We show that the Gauss composition of the first two equals the Levi--Civita midpoint if and only if the latter is invariant under one step of the iteration. We give a sufficient condition for this invariance in terms of an isometry which acts as the point reflection about the Levi--Civita midpoint and exchanges the connection with its dual. Under this invariance, the iterations converge quadratically for sufficiently close initial pairs. Nakamura's iteration is an example of this criterion. The criterion also applies to dual pairs of matrix power means whose Gauss composition is the geometric matrix mean. We give a dually flat Hessian example which shows that duality alone does not suffice. We also give non-flat examples with a Euclidean metric and a parallel cubic form in every dimension larger than one. In dimension one, we characterize the invariance completely by using the Matkowski--Sutô equation. For Euclidean dual pairs, we conjecture that the invariance implies that the cubic form is constant. We prove this conjecture in dimension one and for scalar multiples of a constant cubic form.