  Estimating the shape of an elliptical distribution is a fundamental problem
in statistics. One estimator for the shape matrix, Tyler's M-estimator, has
been shown to have many appealing asymptotic properties. It performs well in
numerical experiments and can be quickly computed in practice by a simple
iterative procedure. Despite the many years the estimator has been studied in
the statistics community, there was neither a tight non-asymptotic bound on the
rate of the estimator nor a proof that the iterative procedure converges in
polynomially many steps.
  Here we observe a surprising connection between Tyler's M-estimator and
operator scaling, which has been intensively studied in recent years in part
because of its connections to the Brascamp-Lieb inequality in analysis. We use
this connection, together with novel results on quantum expanders, to show that
Tyler's M-estimator has the optimal rate up to factors logarithmic in the
dimension, and that in the generative model the iterative procedure has a
linear convergence rate even without regularization.
