A general covariance-based optimization framework using orthogonal projections

We present a general framework for the minimization of a function which is parametrized by a set of covariance matrices over a constraint set. Since all covariance matrices have to obey the property of being positive semidefinite, this characteristic has to be reflected in the constraint set. In addition, the sum of all traces of the covariance matrices shall be upper bounded. Using a preconditioned gradient descent algorithm, we derive an orthogonal projection onto this constraint set in an easy to follow monolithic way such that it directly results from the definition of the projection. Interestingly, this projection allows for a descriptive water-spilling interpretation in the style of the well-known water-filling algorithm. Two possible applications are investigated: the sum mean-square-error minimization and the weighted sum-rate maximization for the MIMO broadcast channel. Simulations finally reveal the excellent performance of the proposed framework.