An Explicit Expression for the Newton Direction on the Complex Grassmann Manifold

Several important design problems in signal processing for communications can be cast as optimization problems in which the objective is a function of the subspaces spanned by tall complex matrix variables with orthonormal columns. Such problems can be viewed as optimization problems on the complex Grassmann manifold, and an effective means for performing this optimization is to use a Grassmannian version of Newton's method. To facilitate the implementation of that method, we provide an explicit expression for the Grassmannian Newton direction for an arbitrary twice differentiable function. We also use an example in which the pairwise chordal Frobenius norm between subspaces is to be optimized to outline a systematic procedure for obtaining the Hessian matrix.