  Matrix functions are utilized to rewrite smooth spectral constrained matrix
optimization problems as smooth unconstrained problems over the set of
symmetric matrices which are then solved via the cubic-regularized Newton
method. A second-order chain rule identity for matrix functions is proven to
compute the higher-order derivatives to implement cubic-regularized Newton, and
a new convergence analysis is provided for cubic-regularized Newton for matrix
vector spaces. We demonstrate the applicability of our approach by conducting
numerical experiments on both synthetic and real datasets. In our experiments,
we formulate a new model for estimating fair and robust covariance matrices in
the spirit of the Tyler's M-estimator (TME) model and demonstrate its
advantage.
