Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness (2209.01229v1)

Abstract: 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.