No Dice:A Deterministic Approach to the Cartan Centroid

Given an m-tuple A = (A₁, . . . , A_m) of positive definite matrices, the minimizer

                     argmin _X Σ δ²(A_k, X),

where δ(Y, X) is the natural Riemannian geodesic distance between pd matrices Y and X, may be regarded as the geometric mean GM(A) of those matrices. It is also known as the Cartan centroid of the matrices, and is closely related to the expectation of the stochastic variable that chooses randomly from among the Ak . Here we establish a novel feature of GM(A): it may be approximated by an explicit sequence of steps, each defined as the geometric mean of just two matrices. Among the potential applications of this procedure, we point out that it can deterministically replace the stochastic arguments that have recently been used to prove that GM(·) is monotone with respect to the Loewner partial order.