Optimal mode decomposition for high dimensional systems

In this paper we present a numerical method for solving a certain rank-constrained matrix optimization problem. This problem is of particular interest for applications in which one wishes to compute a low-rank approximation of the dynamics of a linear system of extremely high state dimension. Our results can be interpreted as identifying a low-dimensional subspace of a high dimensional system in which the projected state trajectories of the system can be best characterized. In order to obtain these results, we consider the general problem of minimizing the difference in Frobenius norm between two data matrices, where one of the terms is multiplied by a matrix of restricted rank with identical left and right images. This problem is non-convex but can be solved using a technique based on optimization on the Grassman manifold.