  The theory of geodesic regression aims to find a geodesic curve which is an
optimal fit to a given set of data. In this article we restrict ourselves to
the Riemannian manifold of positive definite operators (matrices) on a Hilbert
space of finite dimension. There is a unique geodesic curve connecting two
positive definite operators, and it is given by the weighted geometric mean.
The function that measures the squared Riemannian metric distance between an
operator and a geodesic curve is not convex nor geodesically convex in the
operators generating the curve. This is a marked difference to the situation in
linear regression. The literature mainly tries to find numerical solutions that
approximate the optimal curve in a single point.
  We suggest to apply a distance measure slightly coarser than the Riemannian
metric. The ensuing control function faithfully identifies geodesic curves, and
it coincides with the standard control function based on the Riemannian metric
for commuting operators. The control function constructed in this way is
geodesically convex. We are therefore able to find a global and uniquely
defined optimal fit to any given set of data. The generators of the geodesic
curve may also be determined as the unique solution to two operator equations.
