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

Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness