Riemannian distances between geometric means (Q2866218)

Let \(\mathbb P\) be the open convex cone of positive definite Hermitian matrices of size \(m\times m\). This \(\mathbb P\) becomes a simply connected complete Riemannian manifold with non-positive sectional curvature if it is equipped with the trace Riemannian metric \(ds=\|A^{-1/2} dA A^{-1/2}\|_2=(\text{tr}(A^{-1}dA)^2)^{1/2}\), where \(\|\cdot\|_2\) denotes the Frobenius \(2\)-norm. Up to reparametrisation, there is a unique geodesic line joining matrices \(A\) and \(B\) in \(\mathbb P\). Furthermore, the geometric mean of \(A\) and \(B\) coincides with their unique metric midpoint.NEWLINENEWLINEThe main aim of the present paper is to find a kind of geometric mean that is sufficiently close to the weighted Karcher mean on \(\mathbb P\). The author first introduces the notion of weighted geometric mean of \(n\) matrices from \(\mathbb P\). Furthermore, convex weighted geometric means are defined. Then the author provides an estimate of the Riemannian distance between the weighted Karcher mean and any convex weighted geometric mean of matrices from \(\mathbb P\). Next, it is shown that a solution to the initial problem is possible for any convex geometric mean that satisfies a Jensen-type inequality for geodesically convex functions varying over an open set of weights in the simplex of positive probability vectors.