  A simple way of obtaining robust estimates of the "center" (or the
"location") and of the "scatter" of a dataset is to use the maximum likelihood
estimate with a class of heavy-tailed distributions, regardless of the "true"
distribution generating the data. We observe that the maximum likelihood
problem for the Cauchy distributions, which have particularly heavy tails, is
geodesically convex and therefore efficiently solvable (Cauchy distributions
are parametrized by the upper half plane, i.e. by the hyperbolic plane).
Moreover, it has an appealing geometrical meaning: the datapoints, living on
the boundary of the hyperbolic plane, are attracting the parameter by unit
forces, and we search the point where these forces are in equilibrium.
  This picture generalizes to several classes of multivariate distributions
with heavy tails, including, in particular, the multivariate Cauchy
distributions. The hyperbolic plane gets replaced by symmetric spaces of
noncompact type. Geodesic convexity gives us an efficient numerical solution of
the maximum likelihood problem for these distribution classes. This can then be
used for robust estimates of location and spread, thanks to the heavy tails of
these distributions.
